Well I lived in Gumbet when my daughter was a baby so I could recommend you look there or maybe the next resort along BItez, which is a bit quieter and has a nicer beach. Although I am not sure there are any AI hotels there.

Do be careful with AI though because unless you pay quite a bit you wont get a good quality hotel.

If you definitely want AI then you could look at WOW Bodrum, which is in Gumbet and is right on the beach.

If you decide to go to Turkey and want some hints on baby stuff just let me know

Thank you for your help Kedi Yavrusu, I'm still scouring the net for a good deal in Turkey but think we may end up there as we have been before so feel a bit more relaxed going with our daughter. I would like to try Egypt but think it will be too hot for lo in June

Yes I think Egypt would be too hot in June.

We went to Akyaka with our daughter and her family a few years ago. My grand daughter was about 15 months at the time. The beach is perfect for toddlers, sand that is perfect for sand castles, the sea is very calm and shallow for a long way out.
I don't know if there are any AI hotels there though, and as Kedi said you have to be careful in Turkey about cheap AIs' because they really can be poor quality.
We stayed at the Yucelen which is right on the beach, and was very good. Very pretty gardens with little streams running through them, and plenty of ducks around that your daughter would probably enjoy feeding!
I see that the Yucelen is number one on Tripadvisor - so probably still good.
Have a good holiday wherever you choose!

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We're about to go to Hisaronu with a 14 month old and have chosen a smallish hotel - 20 rooms - in the hope it won't be too busy. Managed to find somewhere that will rent us a travel cot and high chair if needed.

In my experience, Turkish people love children and will make you feel very welcome wherever you go.