If I were in his shoes after you had checked I would bet this flop with an over pair, TPGK or if I had paired the flop and put you on overcards. I could also see him having suited connectors and pairing the flop with a straight draw. So a, 56s or 67s is plausible.

I still think you are ahead after the turn and I would bet out about half the pot since there are 10 good outs to catch up if he had something like 57s. I woldn't want to check/raise it here since that card might scare him off of betting and he might take a free card and draw out on you.