There is no need to rent a car in Istanbul, there are eternal traffic jams, and in Sultanahmet you cannot drive through many narrow streets at all, there will be stress, not rest. You can walk around Sultanahmet on foot if you like walking and build routes correctly. You can always take a taxi, you don’t have to take buses.
If you want to see Istanbul live, drive along a five-level interchange, merge into one whole with reckless Turkish drivers, find out what a real “traffic jam” is and much more – be sure to take a car for a day. Make a route, there must be a navigator (bring your own). Sultanahmet, Fatih, Laleli and others will not go anywhere, Just ride without leaving your car in the parking lot for a long time. Once I left in the area of the Galata tower, barely finding a place to park, then I was looking for a car for three hours, and I had to drive it to Antalya itself, moreover, the car was personal, it was a little shy. Good luck !