Skyscanner will give you an easy way to quickly identify the cheapest flight in a given time period. Simply click the flight, and follow the easy instructions to book the flight. Note that you'll be taken to the carriers website to book the flight direct with them (or of course you can phone up to book the flight now you have the details). If you need more help on how to use skyscanner fully, have a look at the guide here: Skyscanner - How to get the best deal
Also, it'd be worth looking at pegasus (Pegasus Airlines), they fly from Stansted, and are not included in the Skyscanner search.
P.S. Currently these are the cheapest one way Bodrum-Gatwick flights: May: £43, June: £71, July: £64, August: £104, September: £91, October: £68 (source)