There's a couple of options here.
If you can limit "your children" to a single browser like Chrome you could write a Chrome extension. But this is nothing new and there are already tools available.
If you're looking to create a commercial product, something a bit more generic then you may need to switch that c# to c++.
You'd need to create a WinSock2 layered service provider and add it to the TCP/IP stack and then interrogate the TCP/IP traffic to identify HTTP requests.
There may be other ways, if so, I'm not aware of them.
Unraveling the Mysteries of Writing a Winsock 2 Layered Service Provider.[
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