he cold called on the flop with second pair after you bet out of position. Error 1 on his part. You bet instead of hoping to check raise here. Possible error 1 on your part (but i think thats splitting hairs!) Turn is a pure luckybox card for him
So no, you played this wrong at no point whatsoever, you just got 5 outered. But as said, being first to act i might look for a check raise here