Safari out of date on my iPad. Update ... how?

Safari is an integral element of iOS/iPadOS - and cannot be updated independently of the OS. If your iPad has been updated to the most recent version supported by your iPad, then Safari is as up-to-date as is possible.

However, some websites incorrectly identify your iPadOS Safari User Agent when used in one or other of its browsing modes - of which their are two - these being Desktop and Mobile.

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.

only way to update safari is to update ios on the ipad

mind you poorly programmed websites may mistake mobile safari as an older safari and give you that message, in which case the latests ios version's safari could still make the website give you that message.