Here is an even worse scenario:

Obama hands over this Gulen guy, and Erdogan shows the world he calls shots everywhere, so Erdogan gets this Gulen guy to "admit" the "USA" was trying to overthrow him. Or whatever and Erdogan decides he's just gonna take those nukes and fuck everybody.

Thus, the Russians decide they need to make a move, a big move. They rattle sabres, make plenty of warnings, then decide they need to zap the base.

Obama decides that it's not just an attack on his pals in Turkey and the inevitable death of US personnel at Incirlik is the Russians starting WW3. Russians figure "fuck it", and decide to roll back on the Ukraine, hard. Gaining pan-European sympathy as the only people with the balls to take action against Islamic expansion, and the war being the fault of US meddling in world affairs, they get popular support to just finalize it - eliminate Washington DC.

America goes in Revolution, and one of the factions asks the Russians to strike certain cities which are known to be power centers of the new Jihad, and the "Syrian refugees", are already armed up and fighting. Thus, each faction calls on foreign troops to help, just different groups of foreign troops, for different reasons...