A friend and I once had like a 6 hour layover in Istanbul on the way back to Israel and we had bought tickets on a normal flight (not chartered) so we were the only frum Jews around and we didn’t feel so comfortable davening in the middle of the terminal. We figured that Ben Gurion has a shul so Istanbul Intl must have a mosque and indeed it did. It was almost neitz which I think is also one of the times the muslims pray at so we went in and they were all going in for their morning prayers. We asked a security guard there if it would be ok if we could use the mosque to pray in and he was a little shocked by the request. He was trying to decide whether to give us permission or not and an old man standing next to him who knew English started telling us, “Yes! Yes! Come it’s the time to pray!” The security guard told us he thought it might be a bit safer if we waited till everyone finished and left so I think we just went back to the terminal and davened on the plane but it was a funny experience.
As far as the airline I’ve only flown on it twice and that was a few years back but it was a decent experience. The airplanes are clean enough and the staff is courteous. Though the politics get a little nasty at times between the countries, I think Turks themselves as a people always strive to be more European and less middle eastern (at least the ones at the airport and on the planes dealing with international travelers) and are just fine on the courtesy. One of the airline people even helped me with directions once at the airport and then smiled and told me “G-d bless you.” I would have no problem booking a ticket with them again and almost did for a recent trip. That being said, always watch your back when flying through a Muslim country, or any non-western country if you are dressing as recognizably Jewish. A little bit of situational awareness goes a long way.