Villian was unknown at the time, caller in the middle was a fish. I thought I will get c/jammed quite often on the turn so wasn't surprised. Question is how often if any are we folding this hand? And do we check turn some x%?

FWIW I snapped turn but I'm questioning my "snap"

PokerStars - $4 NL - Holdem - 8 players
Hand converted by PokerTracker 4

BB: $298.16 (VPIP: 25.00, PFR: 5.00, 3Bet Preflop: 0.00, Hands: 40)
UTG: $542.47 (VPIP: 24.19, PFR: 17.74, 3Bet Preflop: 4.76, Hands: 62)
UTG+1: $334.00 (VPIP: 38.46, PFR: 23.08, 3Bet Preflop: 0.00, Hands: 13)
Hero (MP): $484.90
MP+1: $112.04 (VPIP: 33.33, PFR: 25.00, 3Bet Preflop: 0.00, Hands: 12)
CO: $276.27 (VPIP: 42.62, PFR: 4.92, 3Bet Preflop: 2.86, Hands: 62)
BTN: $186.48 (VPIP: 29.79, PFR: 17.02, 3Bet Preflop: 5.88, Hands: 47)
SB: $396.00 (VPIP: 0.00, PFR: 0.00, 3Bet Preflop: -, Hands: 1)

SB posts SB $2.00, BB posts BB $4.00

Pre Flop: (pot: $6.00) Hero has J J

fold, fold, Hero raises to $12.00, fold, fold, BTN calls $12.00, SB raises to $40.00, fold, Hero calls $28.00, fold

Flop: ($96.00, 2 players) 8 2 3
SB bets $52.00, Hero calls $52.00

Turn: ($200.00, 2 players) 4
SB checks, Hero bets $84.00, SB raises to $304.00 and is all-in, Hero calls $220.00

Well this is never TT 99, so your hand obviously becomes a bluff catcher once he shoves. The problem with bet folding here is that you can't easily have QQ+, and I'm not even sure if you can have all of the 33/22 combos. You're probably very near the top of your range. Assuming that is the case, it might be that equilibrium has you checking behind here with the vast majority of your range. Betting for protection makes a ton of sense of course, but if he's checking a lot of his QQ+ then it is not worth it. You also need to consider that you have other hands in your checking behind range that would like to be protected by checking some stronger hands like this. Otherwise villain can just overshove blank rivers with a ton of his range and be balanced.

A side note: the capped nature of your range seems to make it a pretty bad play for villain to check good hands on the turn. I think probably it is optimal for him to be his good hands almost always and only checking his no-equity giveups like T9.

I think check back turn is better. Yes it would suck to get overbet jammed on river but that will happen less often than getting check/shipped on. But at the same it's probably more likely that he goes for check/ship with Ax than QQ+ since a lot of ppl will check behind their draws and mid pocket pairs in your spot. But at the same time ppl are dumb in these spots and like to miss value like it's their job so idk. So bet/snap and check back both seem fine.

Yes, checking back turn some %. I'm probably doing it more often than I bet. As played, yes snap.

Yea, readless against opponent, I try to stay as close to GTO as possible, and if you're betting the turn with JJ, folding to a shove would be ... so as played, I call.

With no info tho, I like checking the turn.

I'm an lollivepoker player but my instinct is to call his shove as played. On a board like this I would expect him to take a bet bet bet line with the majority of hands that beat us. In my experience this is very frequently a hand like kqhh and he's trying to max fold equity.

I think betting turn is fine too. Checking behind just lets him realize whatever equity he has with his over cards for free. I would only support checking behind turn here if you think history suggests that he's likely to call you with a hand like ace high if the draw misses and you fire river.