I want to send an email to American schools asking about the progress of material review.

It's not so official, and people won't refuse you to wait because of an informal email.

Just say:

May be involved, I want to know whether my application has been delivered, I want to know the status of my application, XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX I want to know if there are any missing materials, and I hope to hear from the other party as soon as possible, so you are ready for further plans. Thank you very much or something.

That's it.

Believe me, this kind of email is neither supercilious nor supercilious, which is good enough. I wrote countless emails to Americans one day, including lawyers, SEC, FINRA, CEO, CFO and so on. And I've read all kinds of emails.