Hi OP, I just want to say hang in there and don’t give up hope! My DS started reception last month (also in a private school) and for the first two weeks was settling in horribly and BITING other children (so far far worse than the behaviour you describe). It was absolutely mortifying, made worse when I raised it on mumsnet and a few vile people made out like he was some sort of devil child and that it was our parenting that was the problem.
It turns out, a month later, that it was just a phase. Thankfully, he’s doing absolutely fine at school now and just had problems settling in (which is exactly what we said to the teachers as he displayed the same pattern of behaviour settling into a new pre-school after we moved house when he was almost four.)
Parenting and discipline were not the issue. We didn’t do anything particularly innovative or different with him, just the usual discipline techniques you probably already know about. After a bad day (ie. if he had bitten someone) we would have a serious talk with him, explain why it was unacceptable, remove his favourite toys and let him know he could only get them back after a whole week of good behaviour. And although he kept biting for a couple of weeks, he did eventually stop - but I’m certain that was when he just felt more comfortable at the school and with his new teacher and new friends. And now the worst thing he’s done at school in recent weeks is go down the slide on his front! What a monster! 
I also agree that I would ask the teachers how THEY are managing the behaviour in class - after all that is their job too, it’s not all up to the parent. If your DS is anything like mine, he was just acting up around new and unfamiliar adults, and does not behave that way at home. So it’s not always the right thing for you to discipline him for things he’s done at school. Just carry on showing lots of love and giving plenty of attention. But it would be useful to know what the school is doing. Are they making him miss playtimes or removing other privileges if he is rude for example? Does he get sent to the head teacher?
Is he bored there or perhaps hasn’t made enough friends yet to feel comfortable? Arrange play dates galore if you think that might be it.
Hugs and hope the situation improves. If my DS has stopped biting kids, I’m sure yours will be fine!