So I did an internship at Univeristy of British Columbia, Canada and I know plenty of people who have gone to research internships in US universities.
1. Internships at universities like UCSD/UMich/UTexas etc :
2 methods:
a. manually : You work hard in the field you want to go for an internship and then you can go through websites of various universities, find out professors who are working in you area, gain some knowledge about current work being done there and then send a mail with a nice cover letter to them. If you are lucky, your mail ll reach to the right person at the right time and you might get an internship there.
b. Brute Force : You send mass mails to people working in your area using some technique. Note that It is banned in some places/campuses and you might be in trouble for doing this and I certainly do not recommend it. But It does save a lot of time. ( Message me,If you want to know more about it, of course for educational purposes only :P) and of course, I never did anything of this sort 😉 :P.