Upvote:2
You should use your American passport to enter and exit ROC, and then your Turkish ID to enter and exit TRNC and Turkey.
If not intending to return to the ROC on this visit, then, when entering TRNC, ask for an entry form with an entry stamp (this will also be stamped when exiting for Turkey), and bring it with you next time you do visit the ROC, as proof you left the island.
Alternatively, you might be able to have the TRNC stamp your US passport on entry and exit despite admitting you on the basis of your Turkish ID, if you explain you need it as proof of having left the island. This would be the most convenient thing, and, unlike entering the Island through the TRNC, will not normally cause issues with ROC authorities.