I'm running 21/13 over 77 hands. He's 16/7 over the same number of hands. The one other time I saw him push over a bet he had a flush, so that made folding the turn super easy. Is betting the turn in the first place correct, though?

PokerStars No-Limit Hold'em Tourney, Big Blind is t150 (9 handed) Poker Stars Converter Tool from FlopTurnRiver.com (Format: FCP)

MP3 (t9820)
CO (t9560)
Button (t6900)
SB (t17549)
Hero (t3870)
UTG (t4880)
UTG+1 (t4625)
MP1 (t2140)
MP2 (t5017)

Preflop: Hero is BB with 8 , 8 .
UTG calls t150, 2 folds, MP2 calls t150, 2 folds, Button calls t150, 1 fold, Hero checks.

Flop: (t675) 5 , 6 , 5 (4 players)
Hero bets t350, UTG calls t350, MP2 folds, Button folds.

Turn: (t1375) J (2 players)
Hero bets t700, UTG raises to t4380, Hero folds.

Raise PF, bet more on the flop. He thinks he might have calling odds with 2 overs if the turn blanks for you and he sees a free river card. Check fold the river. Your river bet is probably pointless because any better hand calls and most worse hands fold. The worse hands that call (two overs) may well bet small into you