I also agree that it is worth looking into the ethnicity of your closest DNA matches, and see if you can figure out how you are related to them.

In addition, its important to put together your family tree based on records that you can find, especially for grandparents, great grandparents, and farther back. This would mean finding records from Ottoman and British times (which admittedly, I don't know much about).

The other thing you could try is to do DNA testing on the oldest person or persons on your grandmother's side of the family. The older the person is, the more likely they will have higher amounts of Jewish DNA, and the easier it will be to identify DNA matches. 

Good luck.
--
Michele Lock

Lak/Lok/Liak/Lock and Kalon/Kolon in Zagare/Joniskis/Gruzdziai, Lithuania
Lak/Lok/Liak/Lock in Plunge/Telsiai in Lithuania
Trisinsky/Trushinsky/Sturisky and Leybman in Dotnuva, Lithuania
Olitsky in Alytus, Suwalki, Poland/Lithuania
Gutman/Goodman in Czestochowa, Poland
Lavine/Lev/Lew in Trenton, New Jersey and Lida/Vilna gub., Belarus