Allowing parents to choose the schools their children attend is inherently superior to forcing them to attend government-run schools. Proof of better outcomes is unnecessary. Sure, some charter/private schools are worse than government-run schools. Parents can easily avoid those.
This is a great attitude to make yourself feel better, but it's not terribly productive if you can't produce some science yourself, or if you are, you know, any way inclined towards respecting humans' individual choices. Economists and sociologists, who are much better at science than educators, try to avoid making the mistake of knocking peoples' choices unless they're obviously irrational, because forcing peoples behavior to be a certain way in the name of "science" is an easy way to be disastrously wrong.
It depends on what you think "better" means. If you use "better" to mean "fits what I think ought to happen", then it makes sense to test market outcomes against nonmarket outcomes and decide which is better. If you use "better" to mean "what most people prefer", then the market is, in most cases, the best way we have to determine that, and talking about proof of better outcomes is analogous to asking for empirical proof that empiricism is a better way to find truth.
What I think natrius means, is that the burden of proof is on the anti-choice option. Surely the default should be giving parents a choice on which schools their children attend. In other words, we shouldn't need evidence to show giving parents a choice gives better outcomes because the evidence should be provided that taking away parents choice is better. If there is any doubt then give parents a choice.