Brock

"If this is the case, then why don't we ever see the stop 'n go in practice?"

It's rare but it still occurs. I think it occurs at a low frequency because the main thing that OOP leading OTT accomplishes is folding out air. That isn't necessarily higher EV for the OOP player then checking, allowing the IP player to bet and get x/raised off the air component of their range or bluff OTR, again allowing the opportunity for OOP to bluff-catch or x/raise bluff.

"Is it because if the person betting the turn out of position is capped then we would just take the bluffs we had to continue with and raise them to balance our raises with nut hands?"

I didn't see mention of the stack sizes so there's not enough detail to answer, however sufficient conditions for that line would require IP raising 100% of nut hands dominating other options including calling and allowing OOP to play an additional street with a capped range.

I think it's unlikely that raising OTT is better then calling when OOP has a capped range, when there is a street left to play. I think raising OTT allows OOP to make many straight forward folds, eliminating the potential for OOP making an additional bet OTR with air that would likely have folded to a raise OTT .

"This seems reasonable, but it also seems like if our range is capped then we'll have to face big bets anyways."

Why can't IP use a mixed strategy with nut hands when calling OTT?

@NY
"I'm talking about raising turn, if you get a good hand" Alright, I agree that many strong hands would raise on the turn. I also expect that to be true on the flop as well, so when he does not raise this texture (flop or turn) I'm not expecting him to have main sets or twopairs. However sets and twopair become a much thinner raise on this river where a 4straight is out and a flush is possible, so in general such hands would be much less common for a regulars river raising range. Since those hands are unlikely due to his flop and turn actions and now much less likely to be raised on this river, I think they are one of the last parts of his range I'm concerned about.

"I bet small because I look more like a draw like this, so I can 3 barrel and get called by some JT, T9, JJ,"
This doesn't make sense to me. Lets say you play your draw the exact same and your opponent knew that. What makes you think river calls with any top pair hands would be part of his strategy, especially on this board where many draws were completed. I would expect the opposite.

I think having a bet size of around 80% pot for flop and turn is a better strategy.

Many draws see a river and most have significant equity so by having a large bet size we are reducing how much villain could potentially win if he calls flop and turn and hits river, while also getting him to put more in on streets where he is behind. That makes the situation less favorable for draws in his range. The bet size you used allows him to see a river for much less while having deeper stacks on the river. The further you go in that direction the less of a mistake your opponent is making by calling two streets with low equity hands.

"lose less if I face a raise (what I was expecting from a better hand on many turns)"
I think that is a secondary concern, because often draws will call twice to see a river. If that's the case get as much as possible from those hands. A important point here is to also have sets and better in your range with the same betsize so your opponents can't confidently semi-bluff raise their draws because they would risk having to fold their equity when they face a re-raise. That's more or less how I play these spots and when I face a raise with a weaker part of my value range here (AA) it is a easy fold, because the range that shares my previous actions and continues vs a raise is strong.

"it looks better to me than two almost pot barrells on flop&turn and then being in a weird spot since we are oop."
You'll be in many weird river spots with AA on this flop and turn regardless of your betsize. However with larger flop and turn bets you're causing your opponent to make a larger mistake and consequently win more from draws and pairs that call twice.

You have a bluff catcher vs UTG and need info on his tendencies to call river. If for instance you knew that he opens really wide and frequently bluffs then calling river is great. However I don't think the average player does that, and I'd assume most unknowns have KQ+ very often with this line.

Maybe you had such reads (you didn't tell us though) and we see that UTG took a very strange line, choosing to bet the turn as a pure bluff with no added fold equity from board texture. Additionally he rivered 2nd pair and decided to bet again. Unless he has a read on you he is misplaying his hand because against the average player in your position that takes your line he'll often only get money in against a range better then his hand. Basically he is turning his hand into a bluff. If you take note of that it can help in future hands.

This is a very wet flop, and there are many hands that a average opponent will call on this texture. Hands like Top pairs, pair+GS, overcards+ GS, middle pairs, and flush draws are calling so I think your post flop bets should be larger for value and semi bluffs (This applies on the turn as well) .

In order for betting all streets to be good (ignoring the times he raises), he will have to call with worse. On this board run out I doubt many regulars would. If he doesn't call with less then 2pair then betting the river here seems like a bad idea. I'd be interested to hear others thoughts on betting river though.

In spots similar to this my plan is usually to bet flop and turn for reasons mentioned above and then decide on various rivers. If river is a brick, maybe I'll plan to check call if I think missed draws will bluff, or if many draws improve on the river and villain doesn't turn weak showdown-able hands into bluffs often, then I'll check and fold.

"I need to win 20% of time, the play is looking a bit strange, no turn raise, it looks like a perfect spot to do it (am I wrong?), but maybe he just almost always get that J9... "

Well it's not only J9 you're behind, 6x is now a str8 and a flush is now possible. I haven't run this through flopzilla or stove but after your opponent raises on the river I think this is a very questionable spot to do anything other then fold. When you say "it looks like a perfect spot" why is that? In my mind the default range I give to a unknown reg that calls flop, turn and raises river on this runout improves enough to expect one pair to be second best very often.