I'm a teacher who has taught in lots of different types of schools across the sectors.
Yes, of course bright pupils can get good grades in a state school. But it isn't always as easy depending on the school.
In private, even with class sizes creeping up teachers are contracted to put in extra hours of support in a way that they arent in state. Many still do in state, but some don't.
In private, yes there is violence and fights do happen but the perpetrators can be removed after just a couple of incidents: they can have zero tolerance policies. It is very difficult to remove pupils from state.
Even in private schools that don't select on ability (there are some) and have a full range of abilities including mild-moderate learning difficulties like was imagined for a comprehensive school, they are able to turn away pupils who they can't look after or educate (eg those with severe SEND who should really not be in mainstream education but are due to the shortage of SEND places). Comps can't cope with them either but have to. These pupils need more attention from the teacher, leaving less for the others. No one wins in this scenario but there is a severe shortage of SEND placements and funding.
Private schools are judged on their grades for individual children. If children aren't achieving, their parents complain and other parents don't send kids there. Some state schools will be graded on how many kids get Cs and above so will not be fussed about making sure little Jimmy gets an 8 rather than a 7.
Private schools mostly (although this is changing with the teacher retention crisis) have subject specialists so if your child is gifted at, say, Physics, their Physics teacher can support that. In state schools often pupils are not taught by subject specialists. This is changing though as the private sector is struggling with staff recruitment and retention too.
These things do make a difference. Bullying can happen anywhere in any school. I would say there are four factors to educational success (by which I mean achieving that pupil's best which might be a 9 or might be a 4!)
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The pupil themselves - do they work hard, act on feedback, ask for extra help when they need it.
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Peer environment; this can't be underestimated. If the ethos at a school amongst a friendship group is it isnt cool to work that will hold all pupils back.
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Home support
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School environment (see above,some of this is almost always better in private because they would close otherwise but there are also some great state schools too).
OP, you know your DD, you can do home support. The things you can't control are peer influence and school environment.