Hello Lucas1948,
Thanks for getting back to us with those additional details. We understand the importance of having private browsing working as expected on your iPad, and we'd be happy to help.
So we're on the same page, have you tried to turn off private browsing and then close all tabs? If not, let's try to turn off private browsing, close all tabs on Safari, restart your iPad, then turn on private browsing, and test.
If the issue persists afterwards, let's restore the device to factory settings (without the backup you will lose data so verify that you have a newly set up backup)
How to back up your iPhone, iPad, and iPod touch
Set it up as new and test before restoring to see if the issue continues there, here's how to restore:
Restore your iPhone, iPad, or iPod to factory settings
If the issue persists afterwards, please contact Apple for further assistance. You can reach out to them here:
Contact Apple Support
Thank you for using Apple Support Communities. Have a nice day.