Immigrants did not take well-paying American jobs.
Rather, the amount employers can afford to pay someone to do that job (i.e., the amount of return that job produces for the employer) fell — for a variety of reasons, many the result of technological growth. It fell to the point where native-born Americans were demanding more than the employers could justify. The only people willing to work at the lower wages that would still allow the employer stay in the black are immigrants.
So, yes, immigrants have many jobs that used to be done by native-born Americans, but they did not take them. And kicking out the immigrants will not get those jobs back.