Well iona, firstly, welcome to the Turkey Central community!
I assume you've had a look at this forum:
Renting in Turkey
Just about anything you need to know about renting a place in Turkey is there. But it sounds like you're interested in prices of apartments in Bodrum, there is also information about which websites to visit, and how to interpret them there. We've got a lot of this stuff covered, but I hope you'll post your personal experiences about renting in Bodrum as well!
I've moved this to the Bodrum forum but left a link here, that way you'll be welcomed to the community, and will also be able to get some good replies to your question!
In any case, welcome to our community, we have a lot of helpful members here!