Lately I've been getting into these spots where the flop is like
J108 hhh
Q97 hhh

Against a terrible player, it's not too tough usually just bet/fold.
Against someone I would consider decent, tight, small winning player, I'm having trouble taking an optimal line.

For instance,

Villain: small winning player, tight preflop, fairly passive post, plays very very ABC. ($250)

Hero: Tight Aggressive player. Raise wider in position. Barrels on good boards depending on opponent. (Covers)

Folds to hero.
Raise to 13 with QcQs.

Villain calls in SB.

Flop J108 hhh ($30~)

Hero bets $20.
Villain check/calls.

Turn 7c ($70)

Villain checks. Hero bets $50.

I'm really just concerned that maybe against someone who isn't going to call turn without a Jack and possibly not even without a heart that this is getting too thin.
Unless he holds Ah or Kh, I beat very few hands. I feel like the 7 is one of the worst cards outside of a heart.

Too thin? Or just a standard bet/fold?

I've seen this spot 3 times in the past week where I didn't have a flush card in my pair and felt a bit lost.

I'm okay with the flop bet but there's no way I'm betting that turn. I think you're behind most reasonable calling ranges at that point.

I'd contend that it's a little to thin. imo, at this point you are essentially bluffing because a tight, fairly passive post flop player will rarely call you here with a worse hand. Only one I can think of is a-j or k-j with the a/k of hearts. Even then, he's got a whole lot of outs.
I probably would have checked ott. 7 is a probably the worst card for you besides a heart. I also like the turn check because you can now catch a lot of bluffs otr and also still may be able to get thin value where he calls you with a jack.