The concept of private prisons basically proves why people who don't understand capitalism shouldn't engage in it.

The entire advantage of capitalism over socialism is that customer choice forces you to make better products, and everything else stems from that. If you only have one grocery store, one school system, one health insurer, one employer in town, it doesn't matter how bad they get, you'll be their slave forever. This is particularly true if the person making the decisions at those institutions doesn't profit from them doing well.

In a private prison system, there's no choice, you can't just switch prisons if you don't like the one you're in. This is the worst system of all, as there's still the profit incentive, without competition keeping it in check.

I think you could do a private prison system well. Maybe let the prisoners choose their prison, with major disincentives for prison owners if prisoners escape. Maybe pay them for every day their newly-released inmates stay out of jail, encouraging rehab programs that actually work. Those would align what's good for the owner with what's good for society, which is all that capitalism is ultimately all about.