The Turkish "problem" is longstanding and kind on both sides as well. Many Turks are not that sure if they want to integrate into German society, and many Germans are not sure if they want them to either (though this sentiment is weaker than it once was). That produces a self-reinforcing situation.
Many of the current Turkish residents came during the 1960s as part of a guest-worker program [1], or are descended from those who did. The expectation was that they would be temporary workers (no path to citizenship) and eventually go home again. The program was closed in 1973, and the government spent much of the late '70s and '80s trying to encourage them to leave, more than encouraging them to integrate. There are many kids who've grown up 100% in Germany (some grandkids too) who aren't German citizens, and until very recently, were not encouraged to become citizens. (Germany does not have jus soli citizenship, and before a reform in 1999, it was also very difficult for someone not of German descent to naturalize.)
The fact that many Turks in Germany come from poor and highly conservative regions of Turkey (generally not Istanbul) doesn't help assimilation, of course. But it also doesn't help that the message for decades was that they aren't German and can't become German. And non-integration is complex, too. When Americans think of unintegrated Mexican immigrants, for example, they think first and foremost of Spanish speakers who can't speak English. But almost all the Turks considered unintegrated do speak German. Some (especially younger generations) don't even speak Turkish fluently (or at all), but only German.
Lots of these aren't German-specific problems, though. Since at least the late 19th century, European counties have been based on the ethnic nation-state, and transitioning to some idea that people can change nationalities, or even have more than one (and what that means) is slow and produces opposition.
[1] https://en.wikipedia.org/wiki/Gastarbeiter