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We avoided being charged 40 euros per person to get out there, once we realized that we could take a local Ferry for about one euro. And we even avoided the really crowded beaches (it was Sunday) by going to a private beach. The only problem was that the private beach was still pretty full, there was really NO beach to speak of (picture a four foot wide sand 'sidewalk'). It looks like this is the best beach that Turkey had to offer (multiple sources confirmed it), so my advice . . . stick to Greece for the beach thing.
Also note that this is really not a picture of us at the beach - it's more of the landfill around the old city with huge boulders and crashing waves, which probably would have been preferable to the island beach resort. ;)