This isn't really my area of specialty, but one example might be the Ottoman-Saudi war (1811-1818). Long story short Egyptian troops were sent, at the Ottoman's request, to fight Wahhabi rebels in an area that now forms part of Saudi Arabia.

This isn't quite an example of Ottoman empire infighting because:

• Egypt's leadership at this point was trying to chart a separate course from the Ottoman empire. It wouldn't be long before Egypt was defacto independant after this conflict.

• The core of the the Emirate of Diryah (the Wahhabi state) wasn't actually in the Ottoman empire (as I understand it). Instead Ottoman involvement came when the Emirate started messing around on their periphery.

So in a way the war was about two states on the periphery of the Ottoman empire, one of which happened to be a very close ally rather than a feudal property of the Sultan or similar, fighting on the periphery of the empire. Hope that makes sense and I haven't made any major factual errors.

Shias were persecuted in the Empire, especially after the rise of the Safavids.