I badly need feedback on this. I commonly receive the same criticism: my animations aren't all too great.

The animation looks serviceable to me. The animation is very clear (although adding arms might complicate things) and doesn't look too stiff. I see two points that would improve the animation:

First, the posing is awkward. The way the head and body is placed it looks like Mario either has a straight back or is leaning backwards, which looks odd. He should look like he's leaning forward (unless you're deliberately going for a weird pose, but that would be more suitable to someone like Waluigi).

Second, you could improve it by adding some secondary actions, i.e. things that would move as a consequence of the up and down motion. the hat could bob up and down by a pixel, and you can distort the shape of the body (like a ball's stretch and squash) or make his gut bounce.

I'm going for a light jog like you'd see in Super Mario 3D World. For that reason, I don't want to make him lean forward; how else can I make the pose look more natural?

Well it doesn't have to be a noticeable forward lean, but I think as it is it doesn't look like there's any forward leaning at all. Part of the reason is that the back is basically a straight line, so this might fix itself when you add the arm motions.

I love it

Did the posing issue end up fixing itself? I'll throw in some secondary motions later.

Looks good so far!

I'd suggest you add a little bounce to Mario's hat.

Yeah, it looks good now!

How's this lazy chest turn? Hat bob eventually.

Also, this.

Anyone who's seen its inspiration before probably immediately knows what it is.
It's also supposed to be reminiscent of Super Mario 3, hence the weirdly shaped and positioned foot.

This kind of sprite is way easier to quickly produce.

I feel the walk cycle is a bit jerky. Yes, it may of been like that in SMB3, but that doesn't mean you have to imitate the sprite 100%.

Well... I like it. In any case, one could easily arrange the frames in a 1 - 2 - 3 - 2 format:

The leg movement becomes a bit messed up, though.

Just add a couple frames? I have no clue.

That jump looks pretty good, though.

The legs on the small walk cycle aren't completing a full motion: the front leg is always in front of Mario and the back leg is always behind Mario. That ends up looking like a limp, you basically have half a walk cycle. If you look at your big Mario animation the leg sweeps all the way behind him before moving in front of him again.

This second point I want to make is just a consistency thing which doesn't impact the quality and matters less if you're not planning on using them for games or something like that, but since the sprites are the same style it would be better if Mario was angled the same way for both sizes.

They're actually not supposed to be the same style. I just randomly decided to make a different Mario for no reason. Skidding pose is very weird. Still working on it.

This is supposed to be a "classic" styled walk animation; that's not an excuse for it being bad, however. If it doesn't work, I'll gladly make a proper one.