is checking correct?

william hill 1/2$ 10 handed, tight table (~30% see flop), folded around to pf raises often. no read on villain who joined the table 10 hands ago and made no visible mistake. very passive game (no flop raises with TP)

hero raises utg+1 with TcTh, mp1 cold calls, everybody else folds. 2 players 5.5sb
flop: Kc Jc 4d
hero bets, villain calls
turn: 7c
hero checks, villain checks
river: 8d
hero checks, villain checks

should i bet the turn? thought not since he called pf and flop- only ahead of aq, at, and small pocket pairs although i also have the flush draw
bet the river when he checks the turn?
figuring that i'm ahead of :
aq- 16 hands
small pocket pairs ~20 hands (excluding smallest)
at - 8 hands
behind:
ak (~8 times since he'll reraise)
kj 9
kq 12
qjs 4
jts 4
so ahead of ~44, behind ~40, very close. he didn't raise the flop but still might have a king since very passive game.
any thoughts?

[sean c]

I would have bet the turn people will call a flop bet with any two cards. No reason to think your not ahead at this point and time and no reason to give a free card here.

[peterchi]

I think you're being too narrow in his hand range. All he's done is cold-call pre-flop, and this means almost nothing so far.

Bet the turn, fold to a raise here.

Check-call the river, expecting to lose if he bets, but it's worth it to see what he's cold-calling with.

Even if she's holding a hand like JT or QJ, she may fold when you keep repping the K. She may also have Axs (possible) or QT, or even just the Ac and will fold the river if she misses. There's no reason to stop betting the turn here.