This is primarily trying to answer another question, but it touches on the question you’re curious about:
I’m afraid, though, that I come at this as someone who’s interested in the Early Turkish Republican era/the Late (19th century) Ottoman Empire, so I don’t know much more about this early/middle Ottoman question than I listed in that answer. (Edit: this particularly doesn’t cover at all when the star and crescent became a global symbol—it obviously appears on flags outside the former Ottoman Empire and outside the context of Pan-Turkism on, for instance, the flags of Pakistan, Malaysia, and Mauritania.)
/u/Chamboz and /u/CptBuck might have more to add.
I did a search of
!g site:reddit.com/r/AskHistorians
and found:
Hope that gives a couple good answers!