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Cut by Patricia McCormick
While confined to a mental hospital, thirteen-year-old Callie slowly comes to understand some of the reasons behind her self-mutilation, and gradually starts to get better.
If you like Cut by Patricia McCormick, here are some other books you may like based on shared issues or themes:
Self-mutiliation:

Blade Silver: Color Me Scarred by Melody Carlson
Ruth copes with an abusive situation at home by cutting herself, until her high school counselor helps her get the treatment she needs to start a new life. (Christian fiction author)

Crosses by Shelley Stoehr
Unhappy at home, Nancy and her friend Katie adopt punk lifestyles and find relief in cutting themselves, until Nancy is forced to confront her problems.
Institutions/therapy/suicide attempts:

The Boyfriend List by E. Lockhart
A Seattle fifteen-year-old explains some of the reasons for her recent panic attacks, including breaking up with her boyfriend, losing all her girlfriends, tensions between her performance-artist mother and her father, and more.

The Burn Journals by Brent Runyon
Brent Runyon was fourteen years old when he set himself on fire. In this book he describes that suicide attempt and his recovery over the following year. (autobiography)