It's very difficult to do this without using more advanced hardware.
On my own setup, I can (and do) catch ALL DNS requests made on my LAN, and either FORCE them to resolve locally (on a dnsmasq instance) or FORCE the packets onto alternative name servers (OpenDNS). The point here is that port 53 is caught and dealt with. I can configure my system so even if you try and use alternative name servers, it can still route you onto OpenDNS (or somewhere else, of my choosing).
So that's one point that kids would not be able to get around.
The second issue is proxies. Now the only way round that is a proxy on port 80 and 443, again, forcing all such traffic destined for those ports through the proxy (even having it deal with SSL traffic, which I believe is possible on newer versions of squid) and then locking down ALL other ports. Proxy can be configured to block even more interesting URLs and sites, plus becomes a nice method for logging browsing activity.
Shove all this configuration onto a custom wireless SSID running on a confined VLAN, and you just about have something that would be child proof. Make sure you run managed switches, password protected, and lock them in a cupboard so children can't just plug directly into the network on a spare ethernet port, and unplug cables to use ports on those switches that are actually configured.
This turns into a rabbit hole of issues that need addressed, and I doubt you are going to get any of this on a off-the-shelf consumer router. OpenDNS is a start, sure, but it can be easily circumvented if the setup allows it.