Private charities do just fine.
Here in Tennessee, we have two of the best pediatric hospitals in the world. Both St. Jude and Le Bonheur are operated with private donations, and no child is turned away from treatment due to their parents' inability to pay. Both of these hospitals, and many others like them, are prime examples of how the private sector takes care of business. I myself donate several hundred dollars to each of them every year, and I don't even have children.