The solution most commonly put forth, in my experience, is single-payer, in which the government serves as the sole, nonprofit health insurance provider. People still pay for their own health insurance if they can, but the government is the middleman between them and health care providers, instead of for-profit companies. In some cases, the government will insure that necessary treatment is given without the patient being charged.

There's a big difference between this and the government actually funding all medical care through taxes, which seems to be what you're saying.