There is no simple way to do this. The main problem you will encounter is that each browser on your system will independently connect to a webserver. That's just a straightforward HTTP connection, usually on port 80. The browser will send the URL in an HTTP request, possibly in multiple TCP packets. So, unless you are going to inspect and reassemble those TCP packets, you're not going to get this information. Even if you did, you'd miss out on the URLs of HTTPS fetches (by design).
An easier solution is to set up a proxy, and hope that the webbrowser doesn't bypass it.