If you bet the turn it'd be considered a bluff, not a value bet. 20% pot equity on a bluff on the turn is definitely doable if you're folding out enough of his range, although we probably have slightly less than 20% equity which we can realize without the Kc or 4c .
The J doesn't hit our opponent's range super hard unless he's floating the flop with a lot of backdoor broadways, which he probably isn't. A couple of his flush draws now have second pair but that's mostly it. One more thing to note is that although JJ is considered good to cold call in this spot, most regulars are probably 3-betting it BU vs CO.
Firstly if you're going to bet the flop with this hand, you really want to avoid check/folding the turn on this card. Even vs fit or fold type, we'd likely have the implied odds to carry on with the hand.
I also reckon that we shouldn't have much of a check/raising range on this turn, except vs opponents with high bet vs missed double barrel. The reason is that all of our legitimate value hands on the turn are too vulnerable to be attempting a check/raise. Therefore our turn check/raising range would be very bluffy unless we're willing to risk a check-back with some strong hands on the turn.
The problem with betting the turn, however, is that we don't want to just fold to a raise. If the sizing is huge then folding is definitely an option. The issue is that we don't want to be bet/folding a lot of our strong nut flush draws, especially vs an opponent who has a strong made hand which we can stack with 7 clean outs. However, it's not like we have a ton of weaker draws in our betting range either, so bet/folding this draw becomes an option, but I'm not entirely convinced about it.
In game I would be bet/calling the turn vs most opponents. I'd also be barreling most rivers if I get flat called, but I'd size it for thin value when I've missed, probably around 40% or less. Vs aggressive opponents who hand-read, I would be check/calling some portion of the time. So while check/folding & check/raising are not an option for me, I would probably only be check/calling to exploit good players.
What would you guys do if 2c appeared on the turn?