Can I update Safari on my iPad?

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 iOS/iPadOS Safari User Agent when used in one or other of its browsing modes - of which their are two - these being Desktop and Mobile.

For recent versions of iOS/iPadOS, 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.

Older versions of iOS may not be compatible with recent versions of Apps. Tags appended to your post suggest that you have an iPad2.

iPad2, iPad3 and iPad mini1 can only be updated to iOS 9.3.5 (WiFi Only models) or iOS 9.3.6 (WiFi & Cellular models).

Apple ended update support for these models in September 2016. Your iPad cannot be updated to iOS 10 or later major versions of iOS, as the internal hardware does not meet the minimum technical requirements for new versions of iOS/iPadOS. This limitation cannot be bypassed; these are 32-bit architecture devices, whilst all recent versions of iOS/iPadOS are designed for 64-bit hardware platforms. iPad2, iPad3 and iPad mini1 are obsolete.

You may wish to consider that compatible Apps for your iPad are increasingly difficult to find - as third-party developers are increasingly dropping support for older versions of iOS preceding iOS11/12/13. As older versions of Apps are removed from the Apple App Store, you’ll not be able to (re)download or install them - which may severely limit the usefulness and utility of your iPad. As such, if you want or need to access most current Apps, requiring more recent versions of iOS, you’ll need to consider replacement of your iPad with a newer or current model.

Of current models, even the entry-level iPad8 has specifications and capabilities that will massively outperform you current iPad - and is compatible with the vast majority of current Apps. Other models are more powerful still.

I hope this information provides helpful and succinct clarity as to your available options - and the limitations of your current iPad.