It definitely happens. I can only think of one concrete case off the top of my head, so here:

>The "kids for cash" scandal unfolded in 2008 over judicial kickbacks at the Luzerne County Court of Common Pleas in Wilkes-Barre, Pennsylvania. Two judges, President Judge Mark Ciavarella and Senior Judge Michael Conahan, were accused of accepting money from Robert Mericle, builder of two private, for-profit juvenile facilities, in return for contracting with the facilities and imposing harsh sentences on juveniles brought before their courts to increase the number of inmates in the detention centers.

http://en.wikipedia.org/wiki/Kids_for_cash_scandal

It's probably both factors added on top of each other. There are other countries where politicians want to appear tough on crime, yet they don't come close to the US in incarcerations.

And the bribery definitely happens. As someone else pointed out already, a juvenile prison bribed judges to convict more kids. But I've also read about a state (Arkansas?) that has a contract with a new privatized prison to provide a guaranteed number of inmates for some time into the future. And you can bet they lobby for tougher sentences too.

Tough on crime can lead to terribly injustice, but for-profit prisons are guaranteed to lead to terrible injustice.