They are both common but they have different meanings and usages. I wasn’t sure I understood. Would you please clarify? I don’t trust your answer. Would you please make sure?

Both are used, but have different usages. "Clarify the details" makes perfect sense in both high-register and spoken vernacular English. "Make sure the details" is not grammatically correct in either, as "make sure" must take either a complementary infinitive "I want to make sure to not miss anything in the survey" or a relative clause "I want to make sure that no details are missed." However, "make sure *of* the details" is correct in spoken vernacular. It is, as you suspected, frowned upon in the high register.

I want to clarify the details. I want to make sure of the details. Make sure of does not mean to clarify (which means to make clear, to understand all the details), it means "to make sure something is the way you think it is or remains ". It is a metaphor, not literally English.

i think "clarify" means make sth clear "make sure" means assure.......

Both are common in North America too; you should be fine!