Private prisons are more a symptom then a cause of high incarceration rates. States began using them in response to prison over-population - and the percentage of inmates in private prisons is relatively low (single digits IIRC).

edit: to be clear I am against privatization of prisons because I think the incentives are woefully misaligned, but wanted to point out it's inaccurate to suggest they're the cause of our high incarceration rate