What's the point of free market for insurance?

There are several insurance companies varying in the level of coverage. In Unites States, the insurance companies are greedy, unreliable and far from full coverage. In more civilised countries (i.e. Poland) the insurance companies usually provide almost full coverage, but still there are still many of them with different prices.

Why is there even competition on the insurance market? Why can't there be a government insurance with full transparency and full coverage? No one would get screwed by insurance then, because assuming the government is democratic (utopian assumption, I know) it wouldn't screw its own citizens.

Clarification: I'm not in favor of government insurance, I just want to understand why is it not the case.