It would take time. Government didn't just show up one day in the health care market, it has been weaving its way through the system incrementally. Removing it from HC would take a couple of years, that is, if you didn't want to shock the current system and have problems like the one you mention.
The best place to start is by increasing the number of practicing health care professionals and removing governments financial incentives to universities for limiting their enrollment in medical schools. Since it takes time to train new HC professionals, we should remove the immigration cap for 10-15 years on foreign born HC professionals to allow as many as possible to enter the country and gain citizenship - this also increases the tax base.
Remove all regulation of the insurance companies, drug companies, and Health Savings Accounts, leaving only the protections necessary to prevent them from committing fraud. Make all HSA's completely tax free. Encourage people to use HSA's to pay for out of pocket checkups and preventative care while maintaining a catastrophic care policy from insurance - That will remove the middle man in all but the biggest procedures and help to bring down costs of the most common procedures.
Change the tax structure regarding insurance. Buying it through our employers doesn't allow for mobility but purchasing a single policy can be far more expensive and taxes do play a role in that. Coupling HSA's and personal catastrophic care that doesn't come from your employer would make HC insurance as affordable and portable as car insurance.
Allow insurance companies to compete across state lines. Enact tort reform. Medicare/Medicaid and the prescription drug program (Medicare part D) all need to be drastically reformed but that's for another post.
Only through measures such as these can we bring down the cost of providing Health Care. Government measures are completely unable to bring down the cost of providing care, the best it can do is reduce expenditures on HC by rationing care, reducing services, and pushing cost overruns onto the debt.
