First of all, the name of the Muslim-Turkish state that defeated Byzantine army in the 11th century at Manzikert was Great Seljuk Empire. As its name mentioned, though a relatively short lived one, it was indeed a great empire that extends from Central Asia to Egypt. Their Sultans, most governors and a significant part of their population were Oghuz Turks, also called as Turkmens-same people who founded Ottoman Empire.
Secondly, The Empire represented an advanced civilization of its time and its army was not a mere raiding force that only seeks plunder. They were trying to find a new land to settle, and to serve Islam while doing so. For these reasons they wisely chose Anatolia and achieved their goal by implementing a sound grand strategy after years of struggle. That's the main reason behind migration of many tribes of Oghuz Turks, which were not too few in numbers, to Anatolia. As experienced by all other empires, Seljuks eventually had been mixed up with many other people they conquered, though generally in small numbers, including other Turkic people, Arabs, Byzantines, Kurds... and especially Persians. This might had been caused a shift in internal power balance as we know that in their last years the langauge that was spoken among the elite was Persian. Returning to the point, most of them were still considering themselves as Oghuz Turks and were slightly slanted eyed like Asian people. Also there was a population boom in Anatolia following the Mongol invasion of 13th century, bringing many (millions?) people to Anatolia and many of them were Oghuz Turks. There were also other Turkish and smaller number of non Turkish tribes and eventually they were all merged together in one pot.
About genetics, I am not an expert but as far as I know, some genes that cause someone having Asian appearance are not of dominant type. Additionally, environmental factors have some effects over genes (depression for example) to some extent, but I cannot prove whether these factors played a major role on the issue we are discussing or not. Also, until today (especially in Ottoman era) there have been many more mixings with other nations' gene pools including European ones. As a last note, no single race has just one fenotype-it is true both for today and for the past. Even in small portions, there are people in a race that express a different fenotype than remaining majority. This internal differences might also have contributed in appearance of the today's Turks. So, there is little wonder about the appearance of modern day Turkish people.