Elected officials on government salaries passed the laws; government-paid law enforcement officers carried out the arrest; and a government-paid prosecutor and (if an indigent defendant) a public defender ensured that justice was served and the defendant’s due process rights were protected in the process.
If the criminal defendant’s case went all the way to trial, then a government-paid judge carried out the trial in an orderly fashion, and jurors on a meager taxpayer stipend decided the defendant’s fate.
This is how the violent criminal wound up in prison, regardless of whether the prison is publicly- or privately-run.
Sorry but your idea of America as a free-market utopia does not actually correspond with reality, in most cases, and especially this one.
I do not want to move to China, thanks.