OK, looking at your targets closely, what I see is that the bullet is "wiping" on its way through the paper on the high side or high/right side of the hole. See the smudge on the high side or about 2 o' clock position on the hole? That tells me that the bottom of the target paper is angled back & away from the shooter, or the bullet is entering on a high, downwards trajectory.

As for twist rate: I believe (but I'm not certain) that those little S&W J-frames in .357 and .38 have a 1-in-18.75" twist rate. They're supposed to shoot everything from 125's to 158's and be able to stabilize them.

First thing I'd try: Put the paper on some sort of backing, like a piece of cardboard or light plywood, so the paper doesn't flap around.

Second thing I'd try is to put some sort of backing in back of the target that could catch the bullet without mangling it. A big wad of synthetic fleece might work for this job. It might be useful to see if the bullet is somehow deformed, or the jacket detached, before we wonder whether the round is tumbling.

Lastly, if you're still getting these sorts of holes when the target paper has a backing, I'd take a look down the bore. Unload the revolver, keep the cylinder out of it, and maybe put a penlight at the  breech while you look down the muzzle, then flip the barrel around and put the light source at the muzzle and look in the breech. Look for any imperfections, dings, burrs, etc. With a 2"- barrel, you should be able to see everything in that barrel without needing a bore scope.

If you have more questions, feel free to flip me an email to the address attached to my account here.