This isn't an exact answer, but this earlier answer to a similar question (along with the clarifying answers after it) goes a long way to help.

Credit to /u/redwashing

This isn't really a unique story. How did South America become Christian so quickly? How did North America? A combination of mass migration, repression, and a confident culture that was able to mobilize enthusiasm for the Turk's faith.

Firstly, I challenge one assertion. The change did not happen quickly. The Seljuk Turks were raiding Armenia and sacking Van in the 1040s. The Seljuks defeated the Byzantines at Manzikert in 1071. The Byzantines retook western Anatolia in the First Crusade, and only forced into full retreat at Myriokephalon in 1176. Anatolia became de jure Muslim in the 1300s, and it took until 1453 for Constantinople to fall to the Ottoman Turks. It took until 1920 for Anatolia to become truly Turkish and wholly Muslim.

Secondly, we have to throw off modern conceptions of religion. Few assertive faiths expand nonviolently, not during eras of military rule and tribal invasion. Generally, peasants are pressured into adopting the faith of their masters, and Anatolia's masters were the Seljuk Turks, who by all accounts were both brutal to their enemies, somewhat generous to their allies, and very infatuated with their newfound Muslim faith. While atrocity stories should be taken with a grain of salt, the annalists of both the Crusaders and Byzantium also report massive violence akin to ethnic cleansing. Most cities were sacked at least once, and the surviving Christians were generally forced into a lower social strata. Again, none of this is unique to the Seljuks. The Byzantines themselves were brutal to their enemies, as were the crusaders and Spanish conquistadors.

The Ottoman millet system which allowed Christian and Jewish self-governance was only in its infancy during the Muslim conquest of Anatolia, and seems generally to have been adopted more to manage the Balkan and Coptic regions. Much of Anatolia was Islamicized before the millet, during the Seljuk and ghazi periods (1000-1300). Moreover, the Kurdish population was largely Muslim already, as attested by historical figures such as Saladin. The Georgians' military strength was humbled by the Mongol invasions; their population largely stayed up in the Caucasus. The process of religious conversion for initially Greek or Armenian populations was likely repressive, with the local elite either killed or taxed into submission. The inability of the Orthodox priesthood to maintain itself in Seljuk Anatolia is relatively well-attested in the Byzantine chronicles. That is the likely crux of the matter. In a vacuum, the peasantry mass gradually adopted Islam under the influence of many traveling Sunni and Sufi clerics.

This was always strange to me. How Did Turks just come and convert whole of Anatolia. And how did Byzantine not put up any big fight. When you look at Byzantine battles vs turks it was usually some small not large scale battles. Guess by the time Turks came Anatolia wasn't much Greek

"The Arab conquests of the Middle East were comparatively swift in contrast to Byzantine Anatolia, where full Islamization took another 400 years from the Turkish conquests in the eleventh century, as the region – with various mountainous areas that sometimes served as natural defenses – was incrementally Islamized and gradually Turkified.30 The invasions of the Seljuq Turks marked a turning point for Anatolia, but for more than a century and a half the Greek populace and the Orthodox church was a significant presence, both because the Seljuk's prioritized stability over total Islamization and because the Byzantine Empire consolidated its political and administrative interesents along the coast of Anatolia an Western Anatolia" Authority and Identity in Medieval Islamic Historiography: Persian Histories ... By Mimi Hanaoka