X (Twitter) does not work with Apple Private Relay turned on when using Safari

If Safari doesn't open a page or work as expected on Mac - Apple Support

Refer to the excerpt Learn more about iCloud Private Relay.

Private Browsing does that for you, but do as you please.

More about Private Browsing: Browse privately in Safari on Mac - Apple Support. As you can see there isn't much about PB that prevents tracking. It essentially limits local storage; that's it.

Lots of people infer things about PB that are not there. IP address tracking prevention and various additional privacy features have been incorporated in Safari for a very long time, so PB itself offers very little additional features.

Learn how the Safari web browser protects your privacy - Apple

Safari Privacy - Apple

Privacy - Features - Apple

Needless to say, if you use Google, don't. Ever. For anything. They lie.

https://spreadprivacy.com/is-private-browsing-really-private/

... etc

Most likely. The bottom line is that some websites simply will not work with iCPR.

"Incognito mode" is also frequently misunderstood. It was probably a bad choice of wording on whoever came up with it, if for no other reason that it does not prevent what you're doing from being observed, tracked, and marketed. Apple calls it "private browsing" which is at least somewhat more accurate. Apple never alleged there was anything "incognito" about it.

By the way it may seem nitpicky but I suggest you don't even use the words "incognito mode" lest you deceive even yourself. It's anything but.

Apple was the first to come up with Private Browsing and they never used any other terminology to describe what it does. Copycats picking up on it started throwing about "incognito mode" to everyone's detriment. Words mean things.