I would definitely just call the check/raise on the flop. Your re-raise on the flop doesn't accomplish too much - even though you're not doing poorly against his range, even a wide get-it-in range is beating you.

Board: Js 9c Th

Hand 0: 36.885% 29.98% 06.91% 184879 42615.50 { QJo }
Hand 1: 63.115% 56.21% 06.91% 346660 42615.50 { JJ-99, KQs, QTs+, J8s+, T9s, 98s, 87s, KQo, QJo, J9o+, T9o, 87o }

This is a very wide range with a lot of pair + draw hands included and gives you 37.5% equity.
If villain continues an even tighter range, you're a lot worse off.

Therefore, I would call the flop to not inflate the pot further. As played, if we give villain a range of { JJ-99, KQs, QJs, J8s+, T9s, T7s, 87s, KQo, QJo, J9o+, T9o, 87o } to shove the turn with, you only have 26.8% equity and will lose an average of 662 chips.
It seems that, unfortunately, you overplayed your hand here and put yourself in a tricky spot.

First off, it's a clear fold pre flop, that deep its not profitable to raise off suited broadways, your range should be more tighter than that, around 15~16%,

As played, like tbettingen said, you overplayed your hand, its ok to just call flop and analyze turn since you're in position and you have advantage agains him so no need to raise his raise, you have top par and OESD, your hand is protected against his strong range.

What you should think before raise is what kind of hands he does that, and if your hand is good against his range, most of times when you dont consider that, you always gonna put yourself in awful spots when you could easily avoid that.

Fold pre unless BB is really bad or the table very weak overall.
Postflop you never really have 3 streets of value, you don't really need any protection, so your hand is a good candidate to check back and probably bet most turns and rivers.
As played, vs a raise, you are quite likely behind already. This flop hits your UTG range well and villain will most likely check/call most of his medium strength hands. You can probably call once and reconsider on the turn. 3betting the flop however is a big mistake and definitely overplaying your hand.