JJ heads up - did I overplay this?

[PokerNoob]

Ok, he caps preflop, so it looks a little better than a random steal. He 3-bets the flop and leads the turn. If he's sane, best case is he's got 15 outs and you're getting 2:1 on turn bets. Is that happening half the time? If he's ahead AND has a spade, you've got one out otherwise two. If he started out with two nice spades, you're scrod. I just don't think you're getting overall value on your turn bets. Call the turn, bet if checked to on the river.