My elder daughter was similar to yours, OP. She was very keen to go to the local comprehensive with her best friend. Best friend was very outgoing and confident. My daughter wasn't, and I didn't think the local school would suit her at all. It had the ofsted rating of 'satisfactory' which is the equivalent of RI today. The ofsted report was full of references to poor behaviour management, and I knew personally two girls who had been taken out of school because of bullying, and knew others. GCSE results were not high at the school (no A level provision) and the quality of music there was not good.
We looked around for alternatives. There was only one other state school school that she could have got into. I thought that was better, but not much better. It was too big (twice the size of the local one), I knew someone who had to withdraw their daughter because of bullying.
In both schools, I thought my daughter would be completely overlooked while teachers concentrated on those who demanded more attention. Neither seemed to have high expectations academically. My daughter hated to stand out, and I knew she would just want to be average and fit in with everyone else. I underachieved at school (a bog standard comprehensive, where it was not cool to be bright) and didn't want to risk that happening to her. I'm not suggesting for a minute that this would have been the case in all state schools, but there are only two in our area that she would have had a chance of getting into.
In contrast, there are a lots of independent schools in our area, so we looked at quite a few. We ruled some out on paper, visited five, and applied for the only two I thought would suit her. Both were academically selective girls day schools. Fortunately she got into our preferred school with a scholarship.
Our experience was great. The class sizes weren't that small (25, but smaller for GCSE years, depending on the subject), but all the girls were bright and behaviour was excellent. I was aware of no instances of bullying the whole time. Expectations and standards were high, both academically and in music, drama and sport, and I was very happy for my daughter to be average.
We worried about the 'poor relation' thing too and, like others, had no issues. A boarding school might have been different (one reason we didn't look at them). My daughter didn't go on the most expensive trips, but neither did the majority of children. I found it refreshing that no-one cared about brands or about having their hair highlighted (unlike at primary school), and that no-one tried to get round the rules about uniform, make up, jewellery etc.
Because the school took girls from a wide geographic area, they put a lot of effort into ensuring a good transition, both in terms of formal activities and in how they divided children between the classes (geographically, so that they travelled with their classmates). The school had its own prep school, and there were a number of other local prep schools who always sent a few children there, but there were also a number of girls, like mine, who were the only child from their primary school.
Of course I will never know what would have happened if my daughter had gone to the local state school. She'd have done ok, I'm sure, and no doubt done well enough at GCSE to enable her to move up to the local 6th form college, but I think given her shyness and lack of self-confidence that either a) she would have dumbed herself down in order to be cool, so she wouldn't have achieved her potential or b) she'd have been teased/bullied/very unhappy.
I also don't think she'd have thrived at some of the other independent schools were looked at. As others have said, it really isn't a question of state vs private. It's a question of school A vs school B.
Someone mentioned bitchiness in girls schools. I can honestly say we didn't see any at my daughter's secondary school. There was quite a bit at her mixed primary school though.