I have been what may be classed by some as a little sad, and to others a little helpful

I have looked into some airlines (not the LLC's, as my system would not allow) ... from AYT (Antalya) to commonly used European destinations for you ... am sure I may have missed alot, but below is what I found to be the best.

AYT-DUS XQ
DUS-LHR BA

AYT-FRA XQ
FRA-LHR BA

AYT-DUS XQ
DUS-MAN LH:BD

AYT-FRA XQ
FRA-LCY CJ:BA

There is one direct flight from AYT-MUC with XQ and also TK

These are the options which came up the most when I searched, most others went throuh IST which is what you wanted to avoid.

Then of course your TATL flight to IAD.

I hope this helps!

Cazzi