If the layover is long enough, don't hesitate, choose Turkish Airlines!
When transiting in Istanbul with Turkish Airlines, as long as the layover is long enough, you can enjoy: a half-day tour, a full-day tour of Istanbul, or hotel accommodation. This is absolutely not a gimmick—the quality is excellent.
This time I flew from Beijing to New York on Turkish Airlines with a 13-hour layover in Istanbul. I signed up for a half-day city tour and saw a number of mosques, churches and other historic sites, took a boat tour along the strait, and it included both breakfast and lunch. The entire trip was on a comfortable deluxe bus.
I didn't get a chance to experience the accommodation or the more extensive full-day tour, but I imagine they are just as good, if not better.
I was originally worried that after a long flight I wouldn't have the energy to go sightseeing, but actually, after the tour, I felt refreshed and much better than if I had slept at the hotel.
Also, this was my first time seeing Turkey, and although the places we visited were tourist hotspots, the civility and friendliness of the people left a deep impression on me.
This counts as a glimpse of Istanbul. — A gift from Turkish Airlines.
Remember to apply for a visa before going. It's an e-visa, completed online, very simple and takes just over ten minutes.