Hi and Welcome to the TLF. I have just had a quick scout about and all I see is Gatwick to Bodrum 27th March earliest flights.
But I do believe Fly Thomas Cook | Cheap Flights - Offers & Deals Online flies all year to Dalaman but dont know about Bodrum. Also some Turkish Airlines fly over winter you may want to try them.
Pegasus do all year round flights to Bodrum but I believe that in the winter months (March) you would still have to change at Istanbul. Maybe you could get a one way ticket with them for the March flight, then book your return with Easyjet/Thomas Cook etc, if you book early enough with Easyjet there flights are quite reasonable and after having used them to go back to the UK a couple of weeks ago, can say that the plane was very comfortable with plenty of leg room.