> The word "exam" isn't really used in American English

Maybe there is some specific narrow regional or other dialect where this is true, but it is ludicrously wrong as a generalization of American English.

Exam's used in American English, plenty. It tends to connote something a bit more serious or formal than a test, but I don't think you'd get much of a difference in reaction just using the two interchangeably, in most contexts. At worst, you'd come off a bit pretentious, using "exam" to describe lesser tests.

"Test" dominates in primary and secondary school, "exam" becoming more common in post-secondary education and for professional certifications et c., but both occur in both contexts.

I grew up in the Cincinnati area and spent a large portion of time in North Carolina. I now live in central Ohio. “Exam” is definitely a word that gets used wherever I’ve lived. Examples have already been brought up — final exams, AP exams, actuarial exams, comprehensive exams, etc.

Well, in the parts of the country I was educated in we had exams, and I would consider the word to be a normal part of people's vocabulary