In the US, students would say SUBMIT in far fewer numbers. Mostly they would say TURN IN.
In other countries it is likely different, but you didn't specify which English speaking country. ;)
In addition, most American students take exams in a classroom not in an exam hall. In a class they would likely have to remain seated until the exam is finished for everyone. An exam hall suggests you can leave. So they would say:
A. "During an exam, I like to turn in my test early because I can't answer much. And then I just sleep."
But even MORE accurately, in such a predicament an American student would say:
B. "During an exam, I can't answer much. So I just sit pretending to answer then turn in my paper when others go up."