AK 4 handed and coldcalling the turn

[Mig]

wouldn't it be preferable to call the turn so SB call +1 BB and then raise the river for +1BB from the BB (total +2 BB more than calling/calling), or is it better to raise the turn and kick SB out of the pot and you only win 1 BB instead of 2 BB (maybe more if SB is dumb enough to call 2 BB on the turn with top pair)

[jrobb83]

I still don't see where I get that much more value out of my hand.

My thinking on the turn was that raising really doesn't gain me anything because I'm not going to get them to but in more bets unless I'm beat. 3-betting likely will net me no more than calling and going for the overcall on the turn, and I don't mind an overcall that much because sb is likely drawing very thin. But when I am behind I am going to get re-raised and have to put in another 2 bets to see a showdown.

Raising the river is correct, but I was seeing monsters under the bed and went for the overcall. But I don't see how this is that huge of a mistake.

For those that say this is horrible: how many bets did I miss and where did I miss them?