Source?

Also what's the proof that private organizations are a better solution to the problem?

Just look at the US prison system.

Not to undermine your point, but you have used the worst possible example. In the top 1000 of worst prisons around the world, most if not all of them are not private. Just take a look a Brazil prison system as en example.

Does Brazil have private prisons that are much better than their public ones?

If not, I don't think you have contradicted his point at all.

There are not too many places that we can look at public vs private institutions for direct comparisons (in the contexts that they operate). Prisons are one, I can think of FedEx/UPS/DHL vs the Post Office as another, and it's difficult for me to come up with much after that.

One glaring problem with private prisons is that it creates an incentive to put more people in jail and keep them there longer.