Is my mom a normal mom?

It sounds like she is a bit immature. You didn't say what you do that gets her so upset, but regardless, getting in your face and yelling and crying is not how a parent should deal with a teenager. I'm not saying that you are innocent in this, because again I don't know what you are doing to get her upset and it may be things that would upset any parent, but she needs to be the parent and dole out consequences if they are needed, instead of yelling and crying about it and acting like a teenager herself.