"Relationship" generally refers to romantic relationships. It would be very unusual to have multiple. You also don't generally "have" a relationship, but rather, are "in" a relationship. The sentence "she has got relationships" is very wrong and unnatural. If the relationship (when referring to people) is non romantic then it needs to be put into a specific context (such as "business relationships") and probably should have some sort of quantifier such as "many" or "multiple" preceding it. Otherwise "connections" may be the better translations.
There's a difference in English between having relationships and being in relationships. I HAVE several different relationships in my life. Some are friendly, some are professional, and one is romantic. I can say that I HAVE a professional relationship with my boss, I HAVE a platonic relationship with my best friend, and I HAVE a romantic relationship with my partner. I have many relationships.
To say that I am IN a relationship means that I am romantically involved with someone. I am IN one relationship, but I HAVE several relationships.