I have good news for you - text search for non-ascii version of latin characters like ü is a long solved problem. That's because even in non-english speaking countries some people don't bother with diacritics, or they don't have a correct layout at that point, or the text is in source code and has to be ASCII, etc.
You can try that now - my browser highlights Türkiye when I search for Turkiye.
You can try that now - my browser highlights Türkiye when I search for Turkiye.