In any man-woman relationship, there is an element of possessiveness that is considered normal. However, in your relationship it does certainly seem that these issues are reaching an unhealthy limit. It is impossible for any relationship to survive in isolation. One must focus energies on other relationships like friends and family and other aspects of one's life like academics and work etc.
Given that your girlfriend seems to experience extreme anxiety when you are not with her and has also tried to harm herself, you must consult a mental health professional immediately. She does seem to need professional help. You must support her and be there for her but these issues cannot be resolved without professional intervention. After a complete assessment, a treatment plan will be formulated that includes, counselling and in some cases medication along with counselling.