KK vs. possible two pair or set

I've been getting burned on overpairs lately, so thanks for advice on this.

With no read, what would you do after this medium size turn bet and why? I ended up min-raising which is a bad play vs potential str8 draws, but I was aiming for a cheap showdown (check behind river or probably fold to a 3 bet). Is this weak or what?

And what if his bet was $7? WIth no read I fold to a full pot bet on this board.

PokerStars No-Limit Hold'em, $0.25 BB (9 handed) FTR converter on zerodivide.cx

UTG ($3.95)
UTG+1 ($52.30)
MP1 ($27.65)
MP2 ($8.10)
MP3 ($14.40)
Hero ($30.15)
Button ($26.05)
SB ($10.35)
BB ($9)

Preflop: Hero is CO with K:spade:, K:diamond:.
1 fold, UTG+1 calls $0.25, 1 fold, MP2 calls $0.25, 1 fold, Hero raises to $1.25, 3 folds, UTG+1 calls $1, MP2 calls $1.

Flop: ($4.10) 5:heart:, J:club:, 8:spade: (3 players)
UTG+1 checks, MP2 checks, Hero bets $1.75 (bet more here?), UTG+1 calls $1.75, MP2 folds.

Turn: ($7.60) T:diamond: (2 players)
UTG+1 bets $4 Hero???
Final Pot: $7.60

Your objective in this hand is to stack AJ/KJ That means you need to build a big pot, preferably before the river (to avoid suckouts). If you want to win a big pot without getting sucked out on (who dosen't), $3 is the minnimum you should be betting on that flop.

As for the turn, you now don't know if their responding to your weak flop bet, thinking you missed, or have hit 2pr+. Leading the turn is a strange line (most monsters would ck-rs) so I'd probably call this down to mimimise losses against a big hand and maximise value from a bluff.

Bet more on the flop

I think I am coming over the top here, I think you are ahead here

I bet the flop 1/2 pot (or meant to...should've been $2+) since there was only a real draw for 9 10, and to minimize a loss to a set, but a weaker bet might mean people think I'm C-betting high cards so it can backfire, and a larger bet will be called both by top pair and sets so I think I see what you're saying. I was playing a little weak after overpair vs. set seemed to happen almost every time the last couple sessions. The turn was strange like you said, why wouldnt he c/r if he had a monster. Thanks for the comments.

He ended calling my min raise and checking the river with a pair of 5 (5 6 suited). Next time I'll get him for more!!!