What type of insurance are you talking about?
All insurance
If you are talking about auto insurance, most states in the US require people to have auto insurance if they want to drive. However, I see a very vigorous, competitive auto insurance market.
It's not much of a marketplace if you have no choice but to pay something to someone. That's called a racket, not a marketplace. It's a hard problem to solve though; roads (or railways) are natural monopolies because of the physical reality of the infrastructure (there's only 1 surface a road can be built on, and the routes are limited by other infrastructure that obstructs those possible routes)
If you are talking about health insurance, that is a big mess in the US. The problem with healthcare in general is that the biggest customers are either chronically ill or dying. Also, in emergency medicine, there is an ethical mandate to treat first. It's not practical or ethical for a healthcare provider to secure means of payment in the middle of an emergency. Most times the indigent or underinsured end up not paying the bill. Therefore, the healthcare providers are forced to raise their prices and try to make up for it by charging more. It just ends up being a feedback loop where the amount the healthcare providers charge keeps rising. In my opinion, the healthcare market is going to end up being socialized no matter what system you try.
Is it practical or ethical to socialize all healthcare, and pay for it with socialised debts that can't be paid at some point in the future? It's easy to say "socialize it to simplify it" now, and forget about how everyone maintains that long term. But remember why we're here on bitcointalk: government debts all over the world have increased more since 2008 than they have since the year 1908. That means big expenses like socialized healthcare are gonna either end or get drastically cut the world over once the government debt market reverses trend.