That is quite strange. I have a vague memory of someone experiencing this same issue in the past. As old school as it seems, one thing to try is to shut off the iPad completely and then turn it on again.
You can do this by holding down the power button for quite a long time then swiping on the button that says "Slide to power off".
Can you please try that and let me know whether it helps?
Also, have you changed anything in the security settings recently? For example, in the Settings app under "Safari" and in the "Privacy and Security" area, is the "Block Cookies" set to something other than "Allow from Websites I Visit"?
