[50NL] A3s...BvB. Bet turn?

Villain is 29/17/2 over 125 hands. Fold blind to steal, 61%.

PokerStars - $0.50 NL (6 max) - Holdem - 6 players
Hand converted by PokerTracker 4

BB: $92.33
UTG: $50.00
MP: $50.00
CO: $48.71
BTN: $51.33
Hero (SB): $50.25

Hero posts SB $0.25, BB posts BB $0.50

Pre Flop: (pot: $0.75) Hero has 3 A

fold, fold, fold, fold, Hero raises to $1.50, BB raises to $4.25, Hero calls $2.75

Flop: ($8.50, 2 players) 3 3 7
Hero checks, BB bets $3.65, Hero raises to $10.75, BB calls $7.10

Turn: ($30.00, 2 players) 9
Hero ???

I don't know whether I should be betting this turn. If I bet it, I think I have to call a jam from villain and I think his range contains so many flushes, he can have over-pairs/77/99 as well, but that is a small % of his range compared to flushes. Is c/c a better option after c/r flop, if I check villain knows that I can hardly ever have a flush here myself.

Thoughts?

Probably fold pre. As played, yah betting turn and jamming river. 3b pot, meh if he has a flush nothing you can do about it. You have enough flushes in your range that it's ok to barrel off here. Especially since any air you might c/r flop with will be betting here.

The preflop call is a little loose, probably a lot loose considering rake and the fact that you probably aren't as comfortable in 3-bet pots oop as you'd like.

It's fine to 4-bet this hand pre but I'd generally lean toward folding it. On the flop I think I prefer check calling down to get max value, but it's probably close with c/r. Obviously on the turn you have an easy bet call, you're doing ok if he has a flush since you have 10 outs. You'll be more than priced in to call it off for the rest even if he tables his flush to you. And there is a shit load of value vs worse hands in his range.

If you check the turn, maybe villain will also go check, so the best option here for me is to make raise to about $15. I think that he has QQ,KK or AA, and to protect your cards from flush make a raise.