Hi Stephen, welcome to the forum.

Glad you joined. You might want to modify your profile to show your general location. That can help out with a lot of things, you may even find a neighbor with a similar problem.
There are several different kinds of thistle, some biennial and some perennial. I don't know of any called American Thistle. Some are worse than others. I'm mostly familiar with Canada thistle which is a perennial that originated in Asia. You might check out musk thistle or bull thistle. It sound like yours could be one of those. Those are biennial, which are actually easier to control than the perennials. Easier doesn't mean easy though.
Where are you trying to control it, a cultivated area, your lawn, or some type of pastureland or hay field? How big of an area are you talking about? That might make a difference in what you can do. They form a big root so if you cut them off they just regrow.
If I catch them young before that root gets ridiculous I generally dig them up and put them where they will dry out. That kills that plant. They are not that hard to control in my garden or landscape beds since I get them young. If the plant is established, usually in the lawn, pasture, or hay field, I use a mattock and cut off the top portion, trying to get some of the root but not trying to dig it all out. So yes it grows back. My goal with those is to never let it go to seed. If it is in a lawn where you mow regularly you can stop it from seeding but those leaves spread out and have sharp thorns, dangerous to kids or anyone else out there. That's why I use a mattock to remove them even in the lawn.
If a flower head ever forms I make sure I get rid of them, usually in my burn barrel. Even if you cut it before the seeds form but after it flowers the plant will continue maturing them as it dries up. It can still make viable seeds. As a minimum send them to the landfill. I never put any part of a thistle in my compost and I don't let any that even look like they are thinking about flowering rot in place. I don't use Glysophate on them but if you spray then early before they start to flower that should kill them. Once they start to flower though it's too late, they will still make viable seeds. I don't know how you feel about chemicals.
One plant can make a lot of seeds. A problem is when a neighbor does not try to control them or when they grow and mature along the side of a road or in fence rows. Once they get established in an area I don't think you ever get rid of them, you just try to control them. My question is this besides roundup could or would putting down grass seed in the yard chock it out ?
Good luck with this and once again