There are lots of lovely resorts around Bodrum. The biggest and livliest is Gumbet but it may be a bit too lively for you, although I'm sure your children would love it. A smaller resort very close by is Bitez, it has a lovely strecth of beach with lots of restaurants and a couple of bars, not much in the way of busy nightlife but fairly similar to Turunc, other smaller but too quiet resorts are Turgutreis or Yalikavak and there also are some resorts more popular with turksih tourists such as Torba, Ortakent or Gumusluk.
Bodrum itself is a lovely town, it doesn't have much of a beach, which is why most people stay at one of the other resorts but they are all close and within easy reach by Dolmus, Gumbet is just 10 minutes, Bitez 15 minutes, the other resorts between 20 and 30 minutes.
If you do fancy trying somewhere on the Bodrum peninsular and want some more info please feel free to ask, i lived there for a few years so I know some of the places very well.
Kedi