Germany, Alemania, Deutschland.
Mexico, Messico, Méjico.
Japan, Nippon, 日本.
China, Zhōngguó, 中国.
Every country gets by getting called different things in different languages. But Turkey insisting on the spelling just because they don't want to be associated to türkiye birds, where the birds are literally named after the country, is a silly move designed to stir nationalistic pride for their near totalitarian government.
The bird is named after a series of misconceptions, with different results in different places.
The country is named after the Turks; you expect it to be the same everywhere.
I'm from a language where you'll never get it right, and most people are pragmatic and don't care about names being the way our mothers do. That's a very Western thing.
Forcing us to try to teach you our language, or half-way butcherings, are considered obnoxious. Go with the English pronunciation. Do anything half-reasonable-sounding, and it's okay too. Don't visibly struggle each time you try to say my name.
For my name specifically, there's also an uncanny valley phenomenon. My name pronounced obviously wrong by an American doesn't raise flags. Americans mispronouncing my native name sounds horribly wrong.