My masks are also made by the pattern on Joann's website. The first dozen or so I sewed on my workhorse Bernina 1230 and had very little trouble with the multiple layers. When I switched to my newer Janome 7700, the difference was like night and day, awful! I second the advice to try a different machine, if you have more than one.

Daffy

I'm using an old Janome with universal 80 needles. I've bent a few, so I get where you're coming from. They're holding up after multiple washes with the ties, not so great with the elastic.

Sometimes it helps to tighten the presser foot tension. It's the metal knob on the top, left of your machine. Righty tighty, lefty loosey.

A "seam jack" or "jeans jack," can help you get over those thick seams. Bernina makes on called a “height compensation tool.” Or you can fold some light-weight cardboard to a thickness of 1/16” to 3/16.” I've used an empty needle pack, too. I have a great tutorial on how to use a seam jack, but I'm not allowed to post the link.

https://cdn.quiltingboard.com/images/smilies/wink.png Here is a link on how to use a seam jack - http://shesasewingmachinemechanic.bl...re-sewing.html I have the one like the pointy one, that comes in handy for turning point, sewing on buttons, making creases or pressing seams

Janey - Neat people never make the exciting discoveries I do.
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How long is your stitch length? Maybe you could lengthen it a bit, like to a 2.5 or even 3.0 when you get to that part of the process. I don't like to pull on the fabric as it comes out the back of the needle, but you might want to tug it a little with a steady pressure to help it along. I've made over 275 masks so far and haven't had much of a problem with my Bernina 570. But I"m still looking for the perfect pattern!

Leslie

I haven't had any problem with my 301 or 404, but if I did, I would move my line of stitching in a bit so I wasn't stitching through the seam allowance along with the mask. I've seem masks with the stitching in about 1/4" - maybe that's why.
Currently, my favorite pattern is the fitted one by Twig and Tale.

I rarely use the hump jumper/ jeans Jack/ etc. for yr's now I simply press down on the front edge of the presser foot, and the machine climbs right over the thick seams. I think it's something to do with putting in contact with the feed dogs. Try it.

Are the pleats stacked on top of each other?

Line up the pleats so they almost touch each other but don't stack.

https://www.youtube.com/watch?v=Bk92ZBt5gQY

This one folds up/folds down.

https://www.youtube.com/watch?v=ceCYbtEHzq0

Using a walking foot is a big help.

The walking foot is the best way to go.