Ok, so there we have a start. Children can be nasty little creatures (never our own, though - of course!) - and so the children with both parents do their best to make the single-parent child feel inferior. Heck, when the mommy and daddy are not particularly happy together, their children are even MORE nasty to the single-parent children. It's not nice, but it's the way children behave in the absence of strong discipline and authority. Which qualities/powers/forces today's schools are conspicuous in their lack of...
For background, feel free to read the book "Lord of the Flies"....
Anyway, there's no point in trying to 'fix' that part of the equation - plenty of perfectly well-adjusted, happy children come from single-parent homes, and deal adequately with our enstupidated schools. It all depends on the child's personality and their position within the school society structure.
It would seem on the face of it that the real problem is your daughter's unrealistic expectations - for herself and yourself, and of what constitutes normality in her world.
However, the 'fix' is relatively easy, I would think.
I suggest that you find something that she enjoys doing - something like dancing, art, horseriding, ice-skating, athletics, windsurfing, swimming, line-dancing - ANYTHING that is outside of what she encounters at her school, and at which she is not already highly proficient. And then encourage her to pursue that activity.
She will make new friends - and being a newbie, she will be the one being cosseted and helped (for a change). When she has attained a certain level of proficiency, she will in turn be able to assist other newcomers - while still being intermediately skilled, as it were. She will become part of a group that is a different group to her school friends - and not feel the need to 'smother' others in order to belong or to show that she is part of the group.
With this will come a new level of confidence and maturity, and her overpowering desire to belong to the school clique will fade away as her behaviour at school returns to normality (or what passes for normality at schools nowadays!).