Why would an Ashkenazi wear a turban except on Purim? Rabbis of Talmud and Koen godol wore it, but Ashkenazim seem to lose the traditional dress and switch to more European one.
Interestingly, Muslims made dhimmis wear a yellow turban or patch, and then Christians retaliated by making Muslims and Jews wear yellow or red patches .. Seems like in places where Jews continued wearing turban, it was enough for Europeans. But Ashkenazim had to wear patches and also pointed hats. Then, in 17th century both non-Jews and Jews in Eastern Europe/Poland wore shtreimels. Later, Poles switch to new French fashions, and shterimels became fully Jewish.
After Khmelnitsky, Jews running away from Ukraine to Germany made shtreimels more popular there. Interestingly, a 1942 The “Jewish Hat” as an Aspect of Social History
by Raphael Straus claims that emphasis on covering head all the time was connected (maybe in some places?) with Shabbatai Tzvi movement