I agree that your daughter is getting overwhelmed by situations. It soulds like she is suffering from social anxiety-- and it is painful for her and inhibiting her from doing things she would like to do. I suggest that you seek professional evaluation so that you can develop a treatment plan that allows her to overcome her anxiety. Since social anxiety can be serious, I would not suggest doing this without professional evaluation.
It sounds like your daughter might sometimes get overwhelmed by situations and not really understand her feelings that trigger the crying. It sounds like she's shy and when the butterflies start to flutter, her reaction is to cry. I know she's only 4 1/2 but have you asked her to use her words to describe to you what she's feeling when she's having these episodes? Try pulling her aside, where it's quiet, where she can't see other people looking at you two, and ask her to descrbe what she feels. It sounds like nerves get the best of her. A lot of emotions are running through her and to her crying is the only solution. Next time you take her to a class, start off by making a firm statement that if she starts crying she's leaving, then do it! Get to the root of why she feels the need to cry. Point out to her she has no reason to cry, mom's not going any where, she's safe, etc. Tell her you only want her to have fun and when mommy sees you crying it makes her think you're not having any fun. If she can't put words to her feelings try to help her by describing how you feel when you get nervous, the butterflies, the excitment, maybe tell a story about when you were younger and nervous. I'd also like to suggest you taking a mommy and me class together, whether it's dancing or swimming or arts class. It might help her to have you right there next to her. Or maybe get her in a playgroup, so she's around more children but the main activity is just free play because it sounds like it's the group that sets her off, not the learning part of a class.