Sort of depends on where you are and what you are looking for.
Id look first at Sahadi on Atlantic Ave., with a huge range of middle eastern mediterranean and other specialty foods, including olives, nuts, feta cheeses, frozen filo pastry, olive oils, spices etc. Good turnover and price.
For Turkish products specifically, there is Birlik at the corner of 60th St and 8th Ave in Sunset Park. They have an in-house butcher as well.
In Bay Ridge there are two areas to cruise - first, there are some greek stores on 4th Ave. s. of 86th St and on Third Ave in the same area. Second, on 5th Ave beginning a few blocks north of there and running up to say, 70th street there are numbers of stores selling middle eastern products, butchers, bakeries, etc. as well as one greek bakery.