Turkey is actually a modern emancipated country when it comes to the cities, and the women (whether Turkish or foreign) wear bikinis whenever they want to.
(Although this might be scrutinized under its current leadership.)
In countryside villages, people are more religious and conservative, so the picture there is different and less exposed (in all its meanings).
The weird global dichotomy is that in other countries — Belgium is a typical example — the Turkish communities tend to be more conservative than in the home hub.
My guess is that the sheer contrast with the indigenous people is too big to overcome in and through the first generations, and as a byproduct especially religiously-related habits tend to surface to mark the difference.
Usually these differences fade away as time slowly passes (and races mix), but when communities with diverging backgrounds meet at first, patience is never around.
In general though, bikinis are definitely allowed in Turkey, every which way —
And worn very well by the locals, if I may say.