Is it normal for my 14yo to get upset anytime she has to do anything without me?

Yes. Most teens begin to pull away from parental involvement. Increasing dependence upon you suggest she is having trouble adjusting to something in her life. How is she doing academically? Socially? Does she feel threatened or bullied? Could she have been assaulted or molested? What are the family dynamics? Many factors need to be considered-make an appointment with her pediatrician today.