My whole approach to this would have been different, I think. You bet the whole pot on the flop - I think I'd probably bet less, like between $10 and $14, to get my information cheaper. In this case he's probably still flat-calling you with the hand he's got.

On the turn then I'd still lead out. The 9 is not obvious trouble unless he has KT. The pot would be in the neighborhood of $44, and I'd bet $20 to see where I stand. At that point he's probably raising; depending on the amount of the raise you can fold, or call (if the pot odds are good) just to see the river and try to get a cheap showdown. On the river I check/fold to a big bet.

The way you actually played it makes for a little trouble. You were aggressive up front, which is fine, but then immediately cooled off on the turn which leaves you in the dark information-wise. A player with any hand or draw is going to bet into you after you check the turn, because it looks like you were bluffing the flop. So you're in the awkward position of not knowing if he still is on a draw, or made a real hand and is ahead of you - and you have to call any reasonable bet for the rest of the hand. It's a perfect setup for him to extract value from his two pair.

I think in order to find out more about what he has, you have to bet something on the turn, which is easier if you lead out for less on the flop (the pot doesn't get huge so fast). Once he hits the 9 he'll probably raise you, and you can put him on two pair or maybe a straight, and play it accordingly. (He wouldn't call the flop and raise the turn with a diamond flush.)