It is normal and there will be times that you just have to let them work it out. But you can teach them how to get along better. For starters, regardless of "who starts it", they both get the same punishment or reprimanding. It takes two to fight, anyone involved is responsible. You have to figure out what that punishment will be, something like a privelidge taken away or making them do an extra chore or something. Whatever works. Maybe you should consider sitting them down together and having a good talk about what it means to be a family and how you expect them to treat each other. You can tell them that no longer will you tolerate them being ugly to one another. Be specific, give examples such as there will be no name-calling, no yelling, no door slamming. Solicit their input on the rules — as well as the consequences when they break them. This teaches kids that they're responsible for their own actions, regardless of the situation or how provoked they felt, and discourages any attempts to negotiate regarding who was "right" or "wrong."
Another thing you can try when they are in a heated arguement is to separate them until they both have calmed. Also, when your own fuse is getting short, consider handing the reins over to the other parent, whose patience may be greater at that moment.
Don't let kids make you think that everything always has to be "fair" and "equal" — sometimes one kid needs more than the other.
Be proactive in giving your kids one-on-one attention directed to their interests and needs. For example, if one likes to go outdoors, take a walk or go to the park. If another child likes to sit and read, make time for that too.
Make sure kids have their own space and time to do their own thing — to play with toys by themselves, to play with friends without a sibling tagging along, or to enjoy activities without having to share 50-50.
Have fun together as a family. Whether you're watching a movie, throwing a ball, or playing a board game, you're establishing a peaceful way for your kids to spend time together and relate to each other. This can help ease tensions between them and also keeps you involved. Since parental attention is something many kids fight over, fun family activities can help reduce conflict.
If your children frequently squabble over the same things (such as video games or dibs on the TV remote), post a schedule showing which child "owns" that item at what times during the week. (But if they keep fighting about it, take the "prize" away altogether.)
Consider establishing a program where the kids earn points toward a fun family-oriented activity when they work together to stop battling.
Good luck and I hope this helps. Another thing, my mom used to make us hug and tell each other we love each other.