No. This is not normal. There is either some information that has been left out or something else going on.
The criteria for baptism includes a well-founded hope that the child will be raised in the Catholic faith. If the priest has reason to believe there may be a problem in this area, he can *delay *(not deny) baptism.
Is she a Catholic that is actively practicing the faith (going to Mass, etc)? Also, if there is a father in the picture, is he Catholic or another religion? Is there any opposition to baptism by the other parent? These things could be factors.
As for the friend who is a priest and is shunning her, that is unusual as well.
What country is this? Perhaps there are some cultural issues at play here as well.