It’s Still in Beta
First and foremost, Private Relay is still in beta. That means that there’s a good chance things won’t work as expected.
Apple is also being quite candid about this, noting that “some websites, networks, or services may need to make updates for Private Relay,” before it will work properly. It’s not just about other sites, however, since as with any beta product, there could be times when Apple is tweaking things that could result in Private Relay failing in various ways.
The bottom line is that if you have Private Relay turned on and you’re having problems accessing certain websites, the first thing you should do is switch it off. On an iPhone or iPad, you can do this in the Settings app by tapping on your name at the top and then going into iCloud > Private Relay (Beta).
On macOS, the option can be found in your Apple ID settings in System Preferences, but don’t just uncheck the box, as that doesn’t always work properly. Instead, click the Options button and choose Turn Off Private Relay.