TPTK Turn & River Play: How many bets you want goin' in?

[Piiop]

I would've played it the same.

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I am almost 100% sure MHIG on the turn,

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How/why?

I wouldn't want to put more than 2 bets in on the turn. With read, I'd consider betfolding a blank river (unless he would also raise river with worse hand) after calling the turn raise. It would prevent a freeshowdown play and get value if he's paired up on a draw and calls. On a scare card, I'd probably check call.