We use Evrensel Sigorta for our house and car insurances, they are reliable and some of their policies are Aviva, so a known company. I know they do health insurance too, they can be found in the building next to Kardz by Karen and are very friendly.
We used to have health insurance, via another agency who worked with Koc and they paid out always.
Check out what your bank offers as well, most of them deal with insurance nowadays.