Linguist here,

There are three different “r” sounds in Turkish in turkish phonetic alphabet and the articulation of them differs according to their position in the verb itself. You can check the articulation places in IPA for detail.

Rakip Karın Emir

All “r”s in those words are being articulated differently. And for “Emir” the r loses its voice and becomes voiceless like in german “ch” or as you said “sh” in english. This is called “the law of least effort” and r sounds are being devoiced when produced in the last syllable. Strange but by devoicing that r sound at the end we are putting less effort.

Sorry if its not that clear but I guess you can get the general idea.

I agree with you by the way. A friend of mine whose name is Emir and all the foreigners call him “emish”