Yes. The owner of an internet connection can absolutely check up on your browsing session if they have the right technical knowledge. Some routers can be set up to remember all URLs a device accesses. So, if you’re trying to fly under your parents’ radar, for example, they could potentially bust you using the family router, even if you’re exclusively using private browsing modes.
For true anonymity, you’ll want to use a VPN. With a VPN connection, all of your internet traffic is encrypted, which gives you far better anonymity and privacy than using private browsing on its own.