There may be some alternative facts that keep your computer hidden But here is what happens with private browsing in Safari: While utilizing Private Browsing, cookies and other files are not saved on your hard drive. Your entire browsing and search history are automatically wiped out.
On a Mac since 1984. Currently: 24" M1 iMac, M2 Pro Mac mini with 27" BenQ monitor, M2 Macbook Air, MacOS 14.x; iPhones, iPods (yes, still) and iPads.
But here is what happens with private browsing in Safari: 