It's a terrible analogy, as carnivals put themselves out there as places that parents can bring kids to enjoy themselves. In general websites are not representing themselves as places to drop off your kids.
A better analogy involves things that are generally intended for adult purposes, as the Internet is. You certainly can operate a business selling construction tools, and generally not be sued when someone lets their kid use one and the kid gets injured. Even if you have knowledge that some customers will buy tools for use by their under age kids, or that they will leave tools accessible to kids, while you're merely assuming that they're supplying appropriate supervision and control.
The fundamental problem with this liability argument that has been put forth is it's based on responsibility laundering. Kids can't judge or consent, yet the argument quietly asserts it's reasonable for their caretakers to be absent in supplying appropriate judgement and supervision as they're interacting with the larger world. The resulting conclusion is that it is everyone else's responsibility to only ever act in ways that will protect those unrestrained kids. The direct implication is that everything accessible on the Internet must be made safe for kids, which is directly opposed to having free and open communications for adults.