Hi! I had a great experience in Istanbul and thought I'd recommend the restaurant I practically lived at! It is called 'Adonin' and is a bar and restaurant in Sultanahmet (disproving the guidebook theories that Sultanhamet is not a good spot for food).
The address is:

Divanyolu cd.
Ticarethane sok.
No: 27/31 34410
Sultanahmet
Istanbul

In easier terms, it is on a street right off the main street and a few minutes walk from the Blue Mosque. It is right near The Ambassador Hotel and Hotel Nomad.

Happy travelling!