UTG: $1.59 (80 bb)
MP: $2.45 (123 bb)
CO: $2.02 (101 bb)
BU (Hero): $4.98 (249 bb)
SB: $0.76 (38 bb)
BB (Villain): $2.53 (127 bb)

Pre-Flop: ($0.03) Hero is BTN with A♥ 4♦
3 players fold, Hero raises to $0.05, 1 fold, BB calls $0.03

Flop: ($0.11) 9♠ A♦ 4♥ (2 players)
BB checks, Hero bets $0.05, BB raises to $0.20, Hero calls $0.15

Turn: ($0.51) 8♦ (2 players)
BB bets $0.59, BU (Hero) folds

I'm a beginner to poker and currently playing 2NL and trying to improve, and these are the types of spots that I'm not at all comfortable at.
When villain check-raises and overbets the turn I don't see them value betting worse hands - is my thinking correct here or could I still be ahead of some of their value range?
If I am indeed either massively behind sets or massively ahead of bluffs, how do I decide if I should call down here?

I think you can probably fold here, tough for V to have many bluffs on that bone dry board, and even if V does takes this line with AJ/AQ sometimes you're still barely getting the odds to call, and you're not going to realize your equity on a lot of rivers anyways. Especially with a bet that big, I think folding is the right play

I would just call down in a BTNvBB pot. Ranges and raising ranges are widest in these spots so BB not only had many bluffs but also Ax hands that are behind 2pr+.

I think for a beginner this spot is pretty neutral

We aren’t gaining much by calling down (which would be by favoured play) but the EV loss of folding is fairly minimal too.

For beginners learning to fold is an important aspect of improving our play. Sometimes we will overdo it and this might be an example.

Until a beginner develops a feel for these spots, tight over folding is less of a mistake than over bluffing or stationing.

I disagree. Overfolding in these spots can lead to some long term bad habits. Blinds vs BTN is the bread & butter of our winrate. Playing these spots correctly is paramount.

We obviously can’t just go folding every tight spot but as a beginning player the first thing to understand is the biggest leak is calling off too many value bets.

Yeah I’d probably peel turn but I do think it’s close. In the NL2z pool it’s a fist pump stack off against some players but a proceed with caution against others. Understanding the difference is why is the vital factor

I'd say here, as we can rule out AA, which villain is probably 3betting pre, villain could have 99 (3 combos), 44 (1 combo), and A9 (6 combos). That isn't very many potential combos that beat you. I'd say you should probably call turn (as my recollection of 2nl is that villains are happy to fire off two barrels as a bluff, but they don't tend to fire off a third too often), and then probably fold if villain jams river. Well, that's what I'd do based on my recollection of 2nl.

I would go ahead and fold the turn. Villain would have to be very creative to be bluffing here. It just looks like A9+ trying to double up.

I tend to agree with this

Need a plan though if we call, if river is a blank how big a bet are we calling, and are we planning to value bet if checked to?