Seat 3 is the button

Seat 1: Player1 ( 560 ) - VPIP: 58, PFR: 8, 3B: 0, AF: 2.8, Hands: 26
Seat 2: Hero ( 480 ) - VPIP: 44, PFR: 27, 3B: 14, AF: 1.7, Hands: 107675
Seat 3: Player3 ( 460 ) - VPIP: 67, PFR: 33, 3B: 0, AF: 1.0, Hands: 3
Player1 posts small blind [10].
Hero posts big blind [20].
** Dealing down cards **
Dealt to Hero [ 9h 7h ]
Player3 raises [40]
Player1 folds
Hero calls [20]
** Dealing Flop ** [ 8h, Jd, 5d ]
Hero checks
Player3 checks
** Dealing Turn ** [ 2h ]
Hero bets [60]
Player3 raises [260]
Hero ?

Standard Defend in the BB. Flop is really good for our range, so a check back isn't out of the norm from our opponent. On the turn, we pick up a very strong draw. I think it makes sense to bet here facing a check back, because we have lots of outs and lots of great rivers we can bluff even if we miss. I think 2/3 pot is a good sizing to get a lot of folds from AX, KX, QX likely on the turn and I think we can even fold some weaker pairs with 2 barrels as well.

The turn raise just makes me hate life. I really don't love any of our options.
We have a strong draw, but the problem is we still only have 9 hi and there's only a handful of weaker draws that we crush. The rest have great equity against us, even random overs is about 50%. Our fold equity on a jam is also likely non existent. The sizing just makes little sense though so I feel his range could be anything here really from value to strong or weakish draws, pair + draw combos and maybe even air. We also block a lot of the draws he could have though.

What is the best play here?

Folding here and losing our equity with such a strong draw is a real crappy spot. But, we are not getting great odds and very few hands don't have good equity against us.

Jamming lets us realize all of our equity even if it's not that great. We would need 40% equity if we get called and our fold equity seems likely non existent in this spot though.

We could also opt for some call + bluff/bluff catch/value bet options depending on the river as well, but the sizing of his raise really throws things off and it just makes for a really odd spot. We need 33% equity on just calling his T260 raise, which we actually have against most hands aside from stronger heart FD's, but calling and folding on a missed draw seems really bad to me though given how much of our stack we will have put in with no chance to win the pot.

Basically have direct odds to call so dont fold. Jamming is worse than calling without fold equity which is nearly non existent here. Just call and imo donk jam when you hit.

You have 15 outs for a straight or flush (also SF) for showdown equity of 15/46 = 32.6%, about what you need to make the call based on pot odds and assuming a hit gives a win. But a 9 high straight or flush is not a sure win.

I did an implied odds analysis where I assumed if you hit, you have an 80% chance to win if villain calls and he will call 75% of the time. The software I used showed you need a river bet of 130 to make the turn call +EV and you get that by going all-in on the river. Without other factors indicating otherwise and assuming the above inputs are reasonable calling the turn bet is correct from an implied odds perspective.

With the math being pretty close, I think you could make a case for folding despite getting the right price for calling. Checking back this flop and then raising with this sizing also leads me to believe he is a weaker player and that we could likely find a better spot.

What ended up happening though is I jammed, because the turn raise sizing just threw me off and I didn't believe it was for value. There just aren't too many hands in his range where he should be strong enough to confidently raise that much on the turn with on this board texture and my thinking was I didn't want to lose out on my equity against something random. At the same time, there are some players who are just betting based on hand strength and over betting their monsters and I could be leveling myself as well in these situations.

Villain ended up having 43o and the river bricked out for both of us. So, it would have actually been a disaster results wise if I had called the turn and check/folded the river. I'm still not entirely sure what the best play is really because I think so much depends on population tendencies with this type of over bet raise and I don't see this line often enough to have a good idea of what it means yet.

calling the turn and check folding the river is still better than folding the turn results wise