As for healthcare, we desperately need less government involvement, not more. We haven't had anything closely resembling a free market in healthcare in the US for nearly a century, so don't try to tell me the market has failed. What we have is instead a series of government enforced cartels. Opaque and incomprehensible pricing structures remove any incentive for patients to shop around. With governments and insurance companies involved in most medical services, there's often little connection between the service received and the direct and visible cost to consumers. Thus, people with poor or no insurance may not get even simple, inexpensive services that might prevent far more costly problems later, while people with "good" insurance may get services that aren't really necessary and don't actually benefit their health.
If we did have a truly free market, it would be entirely possible to set up structures where healthcare providers had incentives to both keep patients healthy and minimize expenses. This isn't some fantastic theory; this actually happened in the early twentieth century. Things only went off the rails in the 1930's, when people advocating greater government involvement and doctors seeking to preserve their prestige collided, only to ultimately team up to plant the seeds of our modern system. As usually happens when you try to mix capitalism and socialism, we ended up with the worst features of both.
Even within our current system, the free market thrives where it's allowed to. Just to give one example, look at cosmetic surgery. Since it's usually considered elective and non-essential, it largely falls outside of the insurance/welfare system. Since people are paying directly out of their own pockets, they have an incentive to seek out the best combination of quality and cost. As a result, providers have an incentive to find ways to provide quality services at reasonable price. As a result, the quality of cosmetic surgery has pretty uniformly improved, while prices have shown little of the explosive growth of the wider medical sector.
It's entirely possible to provide better care for the average person and contain costs, but greater government involvement isn't going to do it. We need drastic reform of professional licensing laws, and Certificate of Need laws should be killed with fire. Health insurance should be restricted to paying for large, unpredictable expenses, while small and predictable expenses should be paid out of pocket. This would both remove some perverse incentives and reduce costs by removing a layer of administration. I'll grudgingly grant the government a role in subsidizing care for people who genuinely can't afford it, but that's about all.