See >>> Browse privately in Safari on iPad - Apple Support
If you are currently using Safari’s Private Browsing mode, to remember the websites that you visit, you must exit the Private Browsing mode.
These instructions relate to iPadOS13/14 (this support page has not been updated for iPadOS15):
Turn Private Browsing on or off on your iPad - Apple Support
For iPadOS 15, the method used to access Private Browsing is slightly different. 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.