It sounds like you could be struggling with claustrophobia if you're finding it hard to breathe when in an elevator. Being surrounded by other people in a confined space is anxiety provoking sometimes. The best thing to do is to focus on your breathing by taking deep breaths and remember that the feeling won't last.
It might be that you're just experiencing some anxiety :) Shortness of breath in elevators is quite normal, and lots of people experience it. The good thing is that elevators are extremely safe, they have to be, they have to meet all sorts of requirements and such. They're safe, and the ride will be over before you know it!!
If you experience panic attacks and anxiety, it could be very possible that you'd experience shortness of breath when in an enclosed space, like an elevator. So yeah, normal. Trying to regulate your breath by deepening it or practicing an on-the-go mindfulness exercise during stressful situations like that could be very efficient in overcoming that specific situation. Hope this helps!
It will normally mean you have an anxiety or a fear of elevators/enclosed spaces an often useful tool is to do 7/11 breathing where you breathe in for 7 seconds and breathe out for 11 seconds. This method will make sure you can relax and think clearly without panicking.