I don't think private schools are necessarily better.
My daughter got a full scholarship at the age of 5 to a new private school. At first we were happy with it and believed that she was doing well and it was much better than the local primary. Unfortunately, we were taken in by the school's lies!
Basically it turned out to be shit academically. Most of the staff were unqualified, the French teacher did not know her un/une or even spell the words correctly. The maths lessons consisted largely of self study from Schofield and Sims book as the class teacher who was also the music teacher was forced to rake in extra money for the school by giving piano lessons!
When my daughter was 11 I gave her a SATS KS2 maths paper and she only got about 3 questions right. She failed to get into a good school at 11, but after attending a proper school for year 7, was very lucky to get a place for year 8 at a well known state grammar school.
The other problem we had as a "charity case" was that we were constantly expected to pay homage to the owner/principle and tell her how wonderful she was. Any criticism/suggestions however mild, were met with stinging abuse. When I complained about my 11 year old daughter being forced to draw a leprechaun for homework (we are Irish and in the context felt it was a gross representation of anti Irish stereotypes) we were shouted at for about 20 minutes by the deputy head. I really wish my daughter had gone to a state primary school!
Moral of the story - coach your kids if they are clever (but still expect to pay up)or move house. I also found that schools change dramatically in just a few years. One school I rejected on the grounds that I went there myself, is the best school around here now!