Time limits in exams were always a source of frustration for me. The time-pressure can end up favouring the students who have a fast, superficial understanding of material, rather than those who are, as you say, more thorough.
This is where test-taking strategies can really help you improve.
The biggest tip I have is to be aware of how much time you have and budget it accordingly. If you find yourself stuck on one problem, skip it and come back to it so that you can spend your time answering the questions you know rather than struggling with what you don't. Try to cover the easiest parts first. This will build up your confidence and give you some "mental momentum."
Another trick is to learn how to get to the point quickly with short answers. Many elemetary teachers will teach students to repeat questions in the answer, which unfortunately can stick with students later in their careers when it's no longer needed. Use point form.