Normal???? I don't think self-harm is "normal" under
any circumstances.
BTW, I really hate the word "normal" outside of statistics and geometry.
It is something that becomes a weird fashion. It can circulate thru a school population like a virus. You have a group of kids who have never done self-harm, one of them starts it, and pretty soon half the female population and a small proportion of the guys have done it. It can vanish as quickly as it started.
During the middle ages, you'd have huge bands of people wandering the land and flagellating themselves for their sins. Pope told them to stop and go home and when they didn't, declared it heresy. However, it never completely died out. Father Junipero Serra, the dude who built a bunch of missions and enslaved the Native Americans out here, would start beating his chest with a stone during sermons. The other friars would have to intervene before he did serious damage. It still happens in some religious displays today. Pain seems to have some weird connection to religious ecstacy.
Flagellant - Wikipedia
You need to learn to forgive yourself. Leave the past behind. Blame and fault finding change nothing and no amount of punishment can rewrite history. Guilt doesn't retract prior actions and cutting yourself doesn't correct mistakes.
Resolving to be the best you can be and not repeat past mistakes is the only way forward. I screwed up, I own it, and I've got to try never to do that again. Live and learn.
If self-harm for the wrongs of the past makes you feel somehow released from responsibility, then it makes it easier for you to do wrong in the future.
If you hurt someone else, genuinely apologize. It does not matter if they accept it. If restitution is possible, do that. If not, find something else that will benefit the world and find a balance that way.
Take out your anger on a punching bag, if you must. It has the advantage of making you stronger and less attractive to a bully. But my coping strategies might not fit your personality.
Most of all I'd suggest talking to a therapist.
Self-injury/cutting - Symptoms and causes