How can I see which URL has been visited with the home network?

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You'll need something running on the router to capture traffic from all connected devices. This functionality often isn't available on ISP provided devices, you'll need a router that either supports this out of the box or one on which you can install alternative firmware such as openwrt or similar. See the Wiki page on Ethernet capture for more info, you are likely to have a switched network.

You could attempt to capture WiFi traffic from the other devices, but that's quite complicated, often not possible on Windows, and can be problematic to configure. See the wiki page on WiFi (WLAN) Capture for more info.