Vinyl distortion - What's the problem?

Sounds like you need to check your cartridge alignment, and your anti-skating setting, and perhaps the azimuth. We don't know what TT or tonearm you have, so it's hard to speculate any further than that.

As TWL says, go over all the basics of the cartridge mounting process. I would add go slow, don't assume it was done right the first time. Make sure your tt is properly levelled. Are you using the brush/stabilizer thing (whatever it is called) on the Shure? You have to adjust the tracking force accordingly. In tracking force, a tiny bit more is better than a tiny bit less. I can't remember if the anti-skating has to be set exactly for the higher setting of the tracking force if you are using the stabilzer/brush thing, check the instructions provided by Shure. Good day.

Sorry I forgot to say the first time that the cartridge is on a brand-new Rega P25 with the stock 600 tonearm. I checked the alignment with the shure alignment card that came with it and set the tracking force for the recommended 1.5 grams with the brush/stabilizer. I don't know what azimuth adjustment is, so I suppose I didn't adjust that. I'll check with a level tonight to see if it is sloping any direction. Looks fine to the eye, but maybe that's not good enough.

One other thing you might check, is to see if your stylus cantilever is bent. You can check this by looking at it directly from the front when it is on the arm rest. It it does not look like it is centered, then it could be bent. This would require a new stylus assembly.

You don't have to worry about adjusting azimuth on the Rega. Make sure the anti-skate is set to the same setting as the tracking force.