Yes. Those are not stinging nettles. Nor do I think the larger plant in the top picture is the same as those in the bottom picture. Top looks like Canada thistle. Bottom not so much.

I sank a narrow shovel close to the base and wiggled it a bit to loosen them and they popped right out. If your soil is more compacted than mine it may take a little more work. I removed over twenty in well under 10 minutes.

Typically there is a deep horizontal rootstock from which vertical shoots grow up. These rather easily pull loose from the main root, may look quite like separate plants but are not. With the result that digging them does not remove the infestation.

So how do I remove the infestation without killing my lawn?

Same as any perennial weed - dig them out. See peren.all's post above. A narrow trowel would do it. Or you spend a couple of dollars on a daisy grubber.

If you want to use herbicides you can spot weed individual plants.

What does a good reliable soil test say about the soil? Canada thistle grows most bestest in soils of low fertility.

kimmq is kimmsr

The lower plant could be Erechtites hieraclifolia. But since it's been mowed it's hard to be sure. Eradication by the same methods.