Throughout most of history, Jews lived significantly better in the Muslim countries than in Christian ones. Moreover, after the expulsion from Spain, many Spanish Jews moved to the Muslim world (from Morocco to the Ottoman Empire). Yet, when Jews were evicted from Central and Eastern Europe, or even after pogroms there (until the 1800s), Jews just moved from country to country in Europe (and sometimes even back to the original country that evicted them). Why didn't they move out to the Muslim world?
I could imagine, it is a kind of spiritual home. Jews immigrated from Muslim countries to Muslim countries (ok, Spain was Christian when the Jews were banished, but it was a new thing, and the Jews remembered on the better days under Muslim authority.)
When Jews were evicted from Central/Eastern Europe they looked for similar societies to settle. They had experience with Christians, so they preferred them. The didn't want to go to complete foreign peoples.
But there also other examples. Jews immigrated to China/Shanghai during the 30's from Germany, Russia and Iraq. Main reason will be, that they had no other possibility.
The question as it was posed is not entirely accurate. The Sephardic Jews are, rightly, the most famous Jewish community of the Ottoman Empire. However, in Istanbul, you could find synagogues and associations belonging to Ashknazi immigrants from Europe. These were all pre-Zionist immigrants from, if memory serves, Russia. In fact, there was a power struggle and conflict in the Jewish community between the European newcomers and the 'native' Sephardic Jews.
This is not to suggest that Jewish immigration to the Ottoman Empire from Europe was large before the late 19th century, but it certainly did exist. I wouldn't be surprised if you found more European Jews in other cities with Jewish populations, such as Izmir, Edirne and, particularly, Salonika.
As to why there wasn't a large-scale immigration, I would offer that European Jews were culturally European and were much more likely to migrate within their cultural world, where their language and practises would have been the norm, than to one which would have been culturally foreign. The same goes for Ottoman Jews, who did not move to Europe in any great numbers during this period because they were more at home with Arabs, Turks, Kurds, Greeks or whichever population they lived amongst.
Eventually both Sephardic and Ashkenazi Jews would eventually emigrate to North and South America, just like Christians and Muslims in Europe and the Middle East.
The history of the Jews post-Diaspora is quite complicated.
For one thing, I believe there was always a significant Jewish minority in the Muslim world, so one answer would be that they did in fact do just what you suggested.
However, there were always some in Europe too. In part, this was because they were inadvertantly encoraged to live there. Christian theology of the Middle Ages prohibited lending money at interest. Thus the only people who could make a living lending money (eg: being bankers) were non-Christians. There's a lot of money to be made in banking, even back then.
This was (sometimes literally) a dual-edged sword though. If you're a ruler who owes a bunch of money you can't repay, one way to get out of it was to get the local citizenry into an uproar about the local Jews, so that they all have to flee for their lives (or stay and get killed). This was the ultimate source of much European anti-semitisim.
We often wonder, how and why is it that our Medieval Ashkenazic ancestors kept on coming back to their destroyed communities to rebuild (much like the residents of Galveston constantly rebuild, after hurricanes and such..).
Why the heck didn't they run away to the Ottoman Empire where the Turks treated Jews relatively nicely? (face it, they needed the Jews, more than the Jews needed them).
Well I don't have the answer to that question, but I can tell you that some Ashkenazim themselves were puzzled by the illogical habitation of Jews in hateful Christian territory. Take a read.
Not all Franco-German Jews suffered from what I once described as a disorder akin to “battered woman’s syndrome”.
As Graetz put it:
When contrasted with the miserable conditions of the Jews in Germany, the lot of those who had taken up their abode in the newly-risen Turkish empire must have seemed unalloyed happiness. Jewish immigrants who had escaped the ceaseless persecutions to which they had been subjected in Germany expressed themselves in terms of rapture over the happy conditions of the Turkish Jews. Unlike their co-coreligionists under Christian rule, they were not compelled to yield up the third part of their fortunes in royal taxes; nor were they in any way hindered in the conduct of business. They were permitted absolute freedom of movement throughout the breadth and length of the empire. They were subject to no sumptuary laws, and were thus able to clothe themselves in silk and gold, if they chose.
Turkey was in short, correctly described by an enthusiastic Jew as a land “in which nothing, absolutely nothing is wanting.” Two young immigrants, Kalman and David, thought that if German Jews realized but a tenth part of the happiness to be found in Turkey, they would brave any hardships to get there. These 2 young men persuaded Isaac Sarfati who had journeyed in Turkey in earlier times, and whose name was by no means unknown in Germany, to write a circular letter to the Jews of the Rhineland, Styria, Moravia, and Hungary, to acquaint them with the happy lot of the Jews under the crescent as compared with the hard fate under the shadow of the cross, and to call upon them to escape from the German house of bondage and emigrate to Turkey. The lights and shadows of his subject could not have been more sharply defined than they are in Zarfati’s letter (written in 1456), whose graphic, often somewhat too artificial language, does not readily lend itself to translation:
“I have heard of the afflictions, more bitter than death, that have befallen our brethren in Germany-of the tyrannical laws, the compulsory baptisms and the banishments. And when they flee from one place, a yet harder fate befalls them in another. I hear an insolent people raising its voice in fury against the faithful; I see its hand uplifted to smite them. On all sides I learn of anguish of soul and torment of body; of daily exactions levied by merciless extortioners. The clergy and the monks, false priests, rise up against the unhappy people of God and say: ‘let us pursue them even unto destruction, let the name of Israel be known no more among men.’ They imagine that their faith is in danger because the Jews in Jerusalem might per-adventure, buy the Church of the Holy Sepulcher (eventually this happened, j.d.). For this reason they have made a law, that every Jew found upon a Christian ship bound for the east shall be flung into the sea. Alas! How evilly are the people of God in Germany entreated; how sadly is their strength departed! They are driven hither and thither, and pursued even unto death. The sword of the oppressor ever hangs over their heads. Brothers and teachers! Friends and acquaintances! I, Isaac Zarfati, from a French stock, born in Germany, where I sat at the feet of my teachers, I proclaim to you that Turkey is a land wherein nothing is lacking. If ye will, all shall yet be well with you. The way to the holy land lied open to you through Turkey. Is it not better for you to live under Moslems than under Christians? Here every man dwells in peace under his vine and fig tree. In Christendom, on the contrary, ye dare not clothe your children in red or in blue, according to your taste, without exposing them to insult and yourselves to extortion; and therefore are ye condemned to go about meanly clad in sad colored raiment (haredim…, j.d.). All your days are full of sorrow, even your Sabbaths and holidays. Strangers enjoy your goods; and therefore of what profit is the wealth of your rich men ( j.d.-הכותב כבר הקדים אתכם, מר גפרסון ומר גון לוק) They hoard it but to their own sorrow, and in one day it is lost to them forever. Ye call you riches your own? Alas! They belong to your oppressors. They bring false accusations against you. They respect neither age nor wisdom; and though they gave you a pledge you sealed sixty fold, yet would they break it. They continually lay double punishments upon you, a death of torment, and confiscation of goods. They prohibit teaching in your schools; they break in upon you during your hours of prayer; and they forbid you to work or conduct your business on Christian feast-days. And now seeing all these things, O Israel, wherefore sleepest thou? Arise, and leave this accursed land forever!” Isaac Sarfati’s appeal induced many Jews to emigrate forthwith to Turkey and Palestine. Their grave demeanor, extreme piety and peculiar apparel at once distinguished them from the Jews of Greece and the Orient, and ere long, the new-comers exercised considerable influence upon the other inhabitants of the countries in which the settled.
But lest one think that conditions for Jews in Judea were utopia:
There were peculiar circumstances connected with the prohibition of the emigration of the Jews to Palestine. The Jewish inhabitants of Jerusalem had obtained permission from a pacha to build a synagogue on one of the slopes of Mount Zion. The site of this synagogue adjoined a piece of land owned by Franciscan monks. The monks raised a clamor, again raising the fear that the Jews would occupy the holy sepulcher..(hmmm..this sounds familiar....-j.d.)
The Pope issued a bull prohibiting any Catholic shipowners (most of whom were conveniently Venetians) to transport Jews to the east.
I am also trying to comprehend an opposite phenomenon, namely that of Sephardim expelled from Spain and Portugal seeking refuge in places like Germany (especially Hamburg), Austria (Vienna mostly) France and even in Eastern Europe (the latter is the subject of my upcoming book). Most of the descendants of these unfortunate souls would eventually meet a violent death on the eastern part of this blood-soaked continent.
For a start, most Jews didn't just move from country to country within Europe. Where the possibility existed, they left Europe altogether. At the end of the 19th century, a veritable flood of immigrants made their way to Germany, sparking fears of Ostjuden - but most of them were actually using Germany as a stepping stone to get to North America (the USA and Canada), to which it was significantly easier to immigrate if coming from a Western European country.
Significant numbers of Russian-speaking Jews (primarily from the Baltic states) made their way to South Africa at this time as well. They sought lucrative employment, freedom from persecution and a better life - the same thing that all immigrants want.
When they couldn't do that, they at least sought something familiar. Why move to North Africa or the Middle East when doing so necessitates learning an entirely new language but doesn't bring with it the possibility of upwards mobility? At times when movement to those regions did bring the possibility of economic advancement (such as after the expulsion from Spain and Portugal), people moved. At times when it didn't, they sought the familiar: other post-Enlightenment Christian countries with a Jewish population possessed of similar traditions and lifestyle customs to their own.
Because the claim Jews "lived significantly better" in Muslim lands is false. There was little difference in the treatment of Jews in Christian or Muslim lands; there were pogroms and periods of tolerance in both.
One thing to remember: Excepting Persia and points east, the "Muslim world" was previously the Christian world. Arabia was peopled by pagan, Christian, and Jewish tribes; Yemen had a Jewish king! The Jewish populations in these lands, North Africa, and Europe were there when Christianity arrived, and still there when Islam arrived. Mostly, people stayed where their ancestors had been, moving only when it was a matter of life or death, and returning when the new place became intolerable.
Actually, there were several Jewish communities throughout the Islamic world.
After the expulsion of the Jews from Spain in 1492, many Jewish communities did emigrate to Muslim North Africa-(specifically, Morocco, Algeria and Tunisia). The Moroccan cities of Casablanca and Fez had sizable Spanish Jewish communities during the Modern Age.
In 1492, the Turkish Muslim Sultanate invited the expelled Spanish Jewish community to reside in various parts of the Ottoman Empire, specifically to Constantinople, Thessaloniki, Izmir/Smyrna, the Aegean island of Rhodes, as well as the Egyptian cities of Alexandria and Cairo. (Although most of the places listed were originally Greek cities and lands, they were under Ottoman Turkish Muslim colonial rule for several centuries).
The Ancient Jewish Diaspora also included pre-Islamic lands and countries, such as Syria, Yemen, Iran and Iraq.
I think one of the reasons was that Europe was more economically developed with higher standards of living. Sometimes Jews knew how to adapt to the circumstances in Europe. For example, in Russian Empire Jews frequently obtained Turkish citizenship so to be counted as foreigners in Russia (and to avoid anti-Jewish legislation which only applied to the subjects of Russian Empire), but remained in Russia.