What to do with medium 2pair vs Turn Re-raise

Pre-flop is a loose defend indeed, but until the turn it’s standard I think.
If you are planning to raise the turn you must fold to a 3bet, because I doubt he would play a weaker 2 pair or top pair like this.

Cheers,
SDK1987

Flop: This flop is very dry and the check from the BTN indicates weekness, that would be confirmed with the fold on the turn.
The SB didn't bet his K on the flop and from his call pre-flop you cannot put him on a hand like top pair.
Your check isn't bad, once you had second pair and had to act before the BTN, who was the agressor.

Turn: This card is a very good card for you, only an 8 would've been better. It's a card that fits the BTN range and with the two pairs you were in good shape to win this pot.

However, SB leads for half-pot and your raise here is not a bad move. BTN folds, so it's just you against SB.
His 3bet indicates a lot of strength, once he leaded the action out of position. From this move, I'd put him on a set.
JJ/KK is not a option here because he flatted pre-flop. Pocket 44/88 is a very possible hand, consistent with is check on such a dry flop, without position and with a BTN that was the agressor.

Top pair here is also a possible hand, but to 3bet that way he would probably need to have AK, that he would've 3bet pre-flop out of position, which he didn't.

Two pairs is also likely, especially J8's. And finally the KJ is also a hand that's consistent with his flop play out of position.

River: The ace seems a card that doesn't change much. The heavy action on the turn tells us that this is not about a draw (a straight in this case), and someone has a made hand.
Is check is strange and he was either trying to make you bet a hand you made with that ace or he got really scared, which is not consistent with is 3bet on the turn. Your action of check behind here could not be better, because he was probably looking for a check-raise spot here.

Conclusion: It's a very tough spot, where you have a strong hand. The BTN let you see a free card, that fit your needs perfectly. I don't blame you for the call to his 3bet, but, in this situation, with the strenght he showed out of position I'd have fold. As I said, I would put him on a set of 44/88 and get the hell out of his way.

Cool, thats very detailled and in-depth! Thank's for spending your time