I'm not sure what you can do other than move schools.
It sounds a lot like my dd's old school. High turnover of staff, poor SATS results and a succession of headteachers brought into turn things around, but none of them stayed so there was no continuity.
She was there from nursery (which was lovely and was what made me decide to keep her at the school) until half way through Y2. It was also our local school and I felt we should support it. Most other middle class parents living very near to the school thought differently though and sent their children elsewhere.
I agonised over my decision the whole time she was there and wondered whether to move her. After all, when everybody else abandons ship you start to wonder why you are staying put. In the end we didn't move her because basically she was happy there and I was happy enough with the teachers she had. We then moved abroad so I was glad I hadn't subjected her to another change of school.
One thing I did do was to get involved with the PTA and I ended up running it for 2 years. If the the new head is supportive then you can achieve a lot. Sadly ours wasn't becase she had too much else on her plate
You could also become a parent governor.