Prison overcrowding bug

I have found that private prisons don’t have any effect on the prison overcrowding situation.

Why should they?

Them are prisons just like the ones state runned, so I think that them should have an impact on that situation.
Unless you consider that no owner of private prisons would help to deal with dangerous criminals and leave all the problem to the state.

Private healthcare has an effect on hospital overcrowding. I see no reason this situation is any different.

Because the funding is completely different. With healthcare, increased demand can actually be met with increased supply when the patients pay for themselves. Will the inmates pay for their own cells when it gets crowded?

Edit: On a second thought, it depends on what you think cranking up the private prison slider means. Does it mean funneling more money into prisons or reallocating from public to private?

Private prisons have similar cost to state prisons so I think that is funneling money into private prisons, not just moving people to private prisons.