I'm not a momma yet, so I can't speak from that experience, but I still jokingly tell my mother that the biggest thing I remember from childhood is my mom saying/yelling "QUIT BICKERING!!!" constantly!

That was her catch phrase. She really DID say it all the time! So, from my own childhood experience I would say the constant bickering and fighting is normal. I remember it so much, that until recently, I was sure that I only wanted one child, just to avoid that constant bickering.
My brother and I were 20 months apart in age and we bickered and fought until we were probably out of high school, even though we still played together, wanted to be around each other most of the time, and really loved each other and were quite affectionate too. He would play Barbies with me and I would build bike ramps with him. We even shared a room until we were around 6 and 8 yrs old.
My mom says that, since all the separating didn't work, she had to just tune it out and ignore it or change the subject and really engage with us, like asking us specific questions that would turn our attention from the pointless bickering. That was easier than fighting with us about it. Sorry, I'm sure that doesn't help, but I just thought I would share my thoughts from being on the other side of it. I so regret now that we made my mom's days so difficult. We had no idea that it was so annoying. We actually didn't even notice when we were doing it.
Beth