Turn off private browsing on iPad

See >>> Turn Private Browsing on or off on your iPad - Apple Support

To access Private Browsing in iPadOS15, it is necessary to first summon the Safari Sidebar using the button at upper left of the screen [1] - and from the menu tap the Private toggle button [2].

To return to regular browsing, tap the sidebar n Tabs button, that appears above the Private button (highlighted here in blue).

Have you tried

Turn Private Browsing on or off on your iPhone or iPod touch - Apple Support