Yes some of the Bosnian Muslim (Bosniak) noble families fled to Turkey. There are still small communities in Turkey that speak Bosnian as a secondary language or even a primary language in their localities. I find their accent of speech most interesting which is ikavian.
However, many of the medieval Bosnians and direct descendants of medieval Bosnians did flee Bosnia upon the Ottoman conquest of Bosnia. Most of the medieval Bosnians were Catholic in faith just like their Croatian brethren in other parts of Croatia except for the majority of the nobility which belonged to a Bosnian Christian sect and there was an Orthodox presence on the eastern frontiers of the Bosnian kingdom which never vanished. Most of the nobility converted to keep their status plus their specific sect of Christianity had no mainstream support from Catholic or Orthodox spheres of Europe. A lot of the Catholics fled for Croatia and Hungary. This is evident in the frequency of “Bosnjak” surnames among Catholic Croats in Croatia and even the “Bosniaks” of Hungary which still exist today but are ethnically Croat and religiously Catholic by virtue of their own self identification as “Bosniak Croats.”
https://bs.wikipedia.org/wiki/Bo%C5%A1njaci_u_Ma%C4%91arskoj (in Bosnian) These were people that fled from Bosnia during the Ottoman days and called Bosniaks to denote what area they come from which was also present in surnames.
If you really look into it its clear that most of the medieval Bosnians were extinguished by the Ottomans and their remnants are scattered or some small elements of them still remain in Bosnia. Medieval Bosnia was sparsely populated land that really had no major centers of administration or organization. In many ways it was a backwater compared to its European neighbors. Ottoman Bosnia was different since the Ottomans immediately set to work on urbanizing Bosnia starting with “Saray-Bosna” which became Sarajevo followed by other towns. Travel, especially for Muslims, was pretty easy within the Ottoman Empire and opportunity in terms of land and titles awaited Muslims which migrated to new parts of the Empire. Bosnia was such a place and the Bosniaks are the descendants of these Muslims who were from all parts of the OE but mostly converted Serbs.