Yet this problem is in a dire need of a free market based solution. Let me explain.
H1B, for those who don't know, is the indentured servitude visa. While you can theoretically move between jobs, your next job has to match the one you had before nearly exactly, and you have to find it _before_ you leave your current job. Once you leave or lose the job you're currently on H1B for, you have to GTFO of the country in _two weeks_. I.e. if your employer finds out you're looking, and they fire you (which they can), you will have no recourse whatsoever. Same with them firing you if you ask for a raise or promo: GTFO.
You're in this limbo state until you get your green card, a lengthy process which for me took 7 years, and for someone from India or China would take even longer.
Companies abuse this power quite heavily by either not promoting/giving raises to H1Bs (because what are they gonna do, quit?) or hiring vastly more experienced workers into entry level jobs, or a combination of the two.
Here's what should be done to fix this:
1. Make the grace period between jobs at least 6 months. If people are abused, they should be free to leave their job without leaving the country, and find their next job over a reasonable period of time.
2. Make it possible for an H1B to move to another job more freely. Any job in the same industry should be acceptable. To make things fair to employers, require repayment of visa related fees if employee moves within the first year.
This alone would fix most of the problems you see on the ground today, including the problem of underemployment of fresh grads. Because movement of H1Bs will be unrestricted, they'll command "market rate". That Eastern European double PhD you hired for your entry level position will be able to move to a job that pays commensurate with her expertise, freeing up the entry level position for a fresh grad. The system will just be much more healthy overall.
However, this will never happen. All those tech CEOs who currently go on ad nauseam about helping people, all of them, without exception, want cheap fresh meat that's not going anywhere. And for as long as H1B existed, they were getting exactly that.