(Apologies - there was a glitch posting an earlier version of this just before I went out for a rather nice dinner.)
Two books by the same Nobel prize-winning author come out on top here. The first, published in 2002, is tagged "Turkey" more often than any other on both Goodreads and LibraryThing, and has the most readers on LT and the second most on GR. The second, published in 1998, is the second most frequently tagged "Turkey" on both systems, and for total readers it narrowly wins on GR and is only a little behind on LT. The two books are:
Snow/Kar, by Orhan Pamuk
My Name is Red/Benim Adım Kırmızı, by Orhan Pamuk
Now, with Turkey there is an unusual problem. Can we count books that are set in the territory of the country we now call Turkey, but are timed before there were any Turks living there? Because if so, there is a clear winner which crushes more recent Nobel laureates (and most other writers) into the dust of ancient battle. It is, of course:
The Iliad, by Homer
I'm also discounting the following novels on the grounds that less than half - sometimes a lot less than half - of the plot (though sometimes a significant section) is set in Turkey. Both Middlesex, by Jeffrey Eugenides and The Historian, by Elizabeth Kostova have more readers on both LT and GR, and GoodReads would add also Inferno, by Dan Brown, Murder on the Orient Express by Agatha Christie, The Sandcastle Girls by Chris Bohjalian and The Forty Rules of Love, by Elif Şafak, though all of them are a long way behind on LT. I am ready to be assured that the last of these is really largely set in Turkey, where it was apparently a mega-best-seller; it wasn't clear to me from online reviews if this is the case.
Otherwise the ancient writer wins for works set in what we now call Turkey, and the Nobel laureate dominates lists of books set in Turkey at a time when it was actually called Turkey.