No Limit Holdem Tournament • 2 Players

$3.40+$0.10

Preflop => super standard min raise KJo @ 19BB
Flop => wet, drawy board
If villain has a wide OOP flatting range he connects with this board a lot and you're probably not achieving anything by cbetting => check it back.
If he is on the tighter side (he's been folding a lot to button opens) he often has overcards or a draw => your cbet decision is dependent on how fit or fold he plays postflop.
I'd check back this flop readless and play the turn in position dependent on how the board runs out relative to villain's assumed flatting range and his bet sizing if he leads into us.
Turn
As played, it is very strange when he flats the flop and then does this gimmicky tiny lead on the turn.
What is he doing this with?
In my experience, gimmicky bet sizing by micro fish tends to be heavily weighted towards their version of creative value lines.
When an overcard to any pair comes on the turn, what would he lead with that he wouldn't have raised the flop with?
I think this lead is usually:
A) A pair that improved to 2 pair on the turn
B) A slow played monster that woke up to suboptimally extract value
C) A weak made hand with some additional equity in the form of a gutshot etc which wants to buy a cheap river card
D) Float which makes weak bluff
You have top pair, but have a relatively weak hand on this board if villain is prepared to put the rest of his money in.
Based on what we narrowed a turn leading range to contain, a raise here will just fold out everything you're ahead of and contract his range to the hands which beat you.
You do best against his whole range by calling his weak lead and making a decision on the river in position depending on how the board runs out and villain's bet sizing.
Subsequent River
If he bombs the river I think you should fold. Most players will not turn a medium strength hand into a bluff so if he bombs the river his range will be polarised between hand types [A+B+(C that got there)] and D. But if he's the sort of player who will make a weak turn bluff, he probably won't then make a large river bluff because it would go against his conservative nature.
If he makes another small bet, you don't need to be ahead much and he can sometimes play Jx this way if he randomly floats the flop with it (maybe he had something like J + a draw on the flop). It will really just come down to the pot odds he lays you compared to how often he has [overvalued medium strength hands + missed draws which are now bluffing].
 
I think you got a little bit too excited about hitting top pair on the turn without considering villain's range relative to the board. You actually had a bluff catcher vs villain's range on this board, not a value hand.