Go to this site, and right Desktop Browsers / Desktop Private Modes to see what does or does not change for various browsers.
@voltronic I really like Librewolf. I do use the update add on, so I get a little notification when it needs to be updated
@haggardmom75
Yes, that extension is essential.
A note for people who use LibreWolf: They disabled Firefox Sync for security reasons, but there is a better alternative for bookmarks sync that is encrypted, cross-platform and cross-browser.
If you want a browser that respects your privacy, I recommend https://librewolf.net/ which is a hardened fork of Firefox.
If you prefer Chromium, Brave is your best bet.
For Android, Bromite is great.