The question asked seems to presuppose that the Ottoman empire was a source of intellectual and technological progress in the Middle Ages and explicitly states that the Christian religion hindered intellectual progress in the West. This looks wrong to me on all counts. Turkish power only rose in the 14th century, at the very end of the Middle Ages, and the zenith of Ottoman power occurred during the Renaissance so Turkish power is not a phenomenon dating from the Middle Ages. Beside, the Ottoman power, though a major geopolitical, economical and cultural power never was (even at its apogee) a major scientific power. Finally, the Church was the only source of intellectual activity during the Middle Ages, so far from hindering intellectual activity, it was the last refuge of intellectuals and of intellectual progress and there was remarkable technological progress in the medieval West (for instance, in the development of mills)
Nevertheless, there are some truths in the question in that the Islamic world did witness a remarkable scientific development during the period 700-1400, with particularly astounding progress in optics, mathematics, medicine and sociology whereas the West knew very little intellectual progress, and the little it knew was mostly in intellectual realms then heavily under the influence of religion (mostly logic and theology), so arguably outside the scope of what we would consider nowadays scientific inquiries. Why is that?
I think the main reasons are economical, sociological and contingent: namely, the West experienced a catastrophic economical decline after the fall of the Western Roman Empire that the East did not suffer, the Islamic world was politically unified whereas the West was in the state of semi-perpetual warfare and the Islamic world was in close interaction with the Eastern Roman Empire and the Indian world (and from there, the Chinese world), three major cultural powers of the time, whereas the western Europe peninsula was relatively geographically isolated. Because of that, the Islamic world had access to Greeks and Romans texts that were all but lost in the Western world and to paper, allowing for the manufacture of books in large quantity. That said, there was a decisive cultural and political factor.
The Abbasid interpreted the Islamic faith as encouraging the quest of knowledge so they had an official policy of pursuing intellectual developments. They invested highly in the establishment of institutions of higher learning (libraries, research center, proto-universities...). For instance, the Caliphs directly impulsed the creation of the library of Baghdad: an institution that became in less than a century the largest library and translation center of the world. The Islamic authorities then considered objective, provable and experimentally sound knowledge to be the most direct reflection of the benevolence of God, and thus especially encouraged the pursuit thereof, in direct contrast with the dominant theology of the Christian church of the time, which valued faith, rejected the physical world as inferior to the spiritual world and emphasized the inherent unknowability of the mind of God. As perhaps the logical conclusion of this general attitude, the Abbasid Caliphate was very open to the incorporation of foreign knowledge and relatively tolerant towards foreign people, independently of their cultural and religious origins. In contrast, one of the greatest intellectual of the Christian Middle Ages, Pierre le Vénérable of Cluny, could read Latin, Greek, Hebrew and Arabic so did translate the Coran and significant part of the Jewish literature, but only for his own benefit and he advised explicitly in Contra Sarracenos (Against the Sarassins) and Adversus judaeorum (Against the Jews) against reading Jewish and Arabic texts, as they were bound to disorient and confuse (we are talking here of a period as late as the mid-12th century).
To conclude, the intellectual development of the Islamic world in the 700-1200 period, compared to the mostly non-existent one of the Christian West during the same period is probably mostly due to economical and socio-geographical factor, but was also the result of a strong official and religious policy of advancement of objective knowledge about the physical world, in direct contrast to the dominant theology of the Church of the same period.
This last paragraph is unrelated to the rest of the answer, but in view of the answers of Pieter Gerkens and Anixx, I think it is necessary to recall that the Islamic world of the 700-1400 did increase knowledge, in fact in stupendous proportions, and did not simply preserve accumulated knowledge. So here follows a short list of scientific achievements of the Islamic world which went far beyond comparable Greek knowledge (and which were typically incorporated in the West only in the 15th century). Most items in this list were completed by 1100, a time by which the scientific achievements of medieval Europe are essentially nil.
Use of the digital system in order to streamline arithmetical operations, and hence first systematic methods of solving algebraic equations (hence of course the words algebra and algorithm).
First solution of all cubic equations with positive roots using conic sections.
Introduction of spherical trigonometry.
First proof that Venus is between the Sun and the Earth.
First correct anatomy of the eye.
Foundations of modern optics, proof that light travels in straight lines and determination of the size of the atmosphere.
Outstanding progress in medicine, especially in the scientific description and isolation of a number of afflictions.
First correct computation of the longitude span of the known world (from the Canaries Island to India).
First attempts at a rigorous formulation of historical and sociological
inquiries.