Could you please describe how it's possible to write about "I always made some mistakes on the exams." with verb "did", if it's really possible. Because, for example, "I always did badly...", in my opinion, it sounds more about some bad actions rather than mistakes on the exam, isn't it?
To do well is a colloquial verb-like phrase that means you got a good outcome from something. Usually it implies that you brought about this outcome through your own actions (you did something, and you did it well enough that you got the results you were hoping for); but the focus is on the outcome, not on your actions themselves.
Equally, to do badly/poorly is the opposite - the outcome from something was not as good as you hoped for.
This phrase is very common when talking about exam results, so "I always do badly in exams" is the natural way of expressing the idea that "in all of my exams I [always seem to] get low marks".
You can still use "I always make some mistakes in exams" if you want to focus on your actions: the wrong answers that you write down and your thoughts at that time. It also suggests (to me at least) that you are aware of the mistakes as you're making them, but you can't or don't avoid them (low on time? Only realise it afterwards? Deliberately do it because you don't want to pass?). On the other hand, "I always do badly" has more of a feeling that you're not sure why, but you always seem to get low marks.