LAX-IST-LAX, then add IST-whereever-IST as separate tickets. TK is good value if you order a couple of months in advance, average around 200-300EUR for a return to almost anywhere on the European continent. The UK is a tad more expensive, but not much.

PS
London and Paris? How about going off the beaten track? Barcelona, Berlin, Brussels and Budapest could be options, and that's just the one's starting with B. The next letter in the alphabet suggests Copenhagen, Clermont-Ferrand and Cologne, and I would very much recommend the former. D suggests Dresden, Dublin and Dubrovnik, etc ad nausem. Plenty to see, even more so if you refrain from stopping at the usual tourist traps.

PPS
You won't see much with a 1-2 day stopover, won't even scratch the surface actually. But if the only thing you're looking for is putting a pin on a map, I suppose it fits the bill.