What causes you to believe that your problem is related to cookies and Safari?
Have you verified similar behaviour when using a different browser, such as Firefox?
Have you updated to the most recent version of iPadOS - this being iPadOS 15.2? The tags appended to your post suggest that your iPad has iPadOS 14.x installed. If you have not already updated to iPadOS 15.2, you would be well advised to do so.
Some websites incorrectly identify your iOS/iPadOS Safari User Agent when used in one or other of its browsing modes - of which there are two - these being Desktop and Mobile. Misidentification of the browser Agent can often manifest as the website reporting that your browser is out of date, or won’t properly render a page.
It’s easy to switch between modes if a website isn’t functioning correctly. From the desired web-page, touch the aA icon at the left of the Address Bar; this will reveal a menu. From the menu select Website Settings, then select (or deselect) Request Desktop Website, then tap Done. Now refresh the webpage.
The chosen setting will be saved for the selected website - and can be simply reversed if needed.