Don't I'm sure its just some type of problem... IDK but i have the same thing, i cut myself, you need to talk to someone you trust like a counselor and let it all out, you have to tell them what you are doing and i'm sure they can figure it out.
Mine started at 10 too but I now know the specific cause and all the deatils of what, why and how etc. It was a fatal car accident which I obviously survived but others didn't.
Finding the cause is always a target for us and whilst it isn't always possible to find such it must be a place to start.
You ask is there any way to stop these thoughts without medical or counselling support. Well, self help books offer such help but if you are suicidal it's unlikely you'd be able to practice what they preach.
Anybody who is suicidal really must be seen by doctors and if you are also self harming the hospital is the place you should be.
Self harm, physical, is really just part of depression you know. Depression itself is self harm as we bombard ourselves with negative thoughts etc about ourselves which harms us deeply. Physical self harm is an extension of that and most common amongst teenagers. Often it is a way to cry out for help. If your parents aren't listening to your words then how will they respond to self harm? Show them and they must respond. Else just go to a hospital emergency ward. Better of course would be if you could go see a doc but that depends on your age doesn't it?
Whatever you do please do not accept the thoughts as anything other than thoughts prompted by feeling extremely bad, don't accept the self harm as that could destroy your future easily and do not accept no medical intervention. Go see the school counsellor if you have one.
And no, it is not normal for teenagers at all. Never was, never will be. That's just avoidance tactics, like "I don't want to talk about it as it's too scary:"
Please dont do anything to harm yourself any further...Your life has just started you have so much ahead of you try to make the best of it...Obviously someone cares about you if they didn't let you jump off a bridge...Youll get through this dont let your parents stop you from getting help...Believe it or not we all have a purpose...
Best wishes