The YouTube App now requires iOS11 or later.
If your iPad is too old to be updated to iOS11, you’ll also find that you’ll also not be able to directly access the YouTube website from your iPad. There is third-party web-portal from which you can continue to access YouTube content:
https://www.invidio.us/
Returning to your iPad...
If your iPad is currently running, or cannot be updated beyond iOS 10.3.3/4, then have an iPad4.
Update support for iPad4 ended September 2017. The final iOS releases for this model were iOS 10.3.3 (WiFi Only models) and iOS 10.3.4 (WiFi & Cellular). iPad4 cannot be updated to any later major version of iOS as it lacks the minimum hardware requirements needed to run more recent versions.
Many App Developers have also dropped support for older devices and iOS versions preceding iOS11/12/13 - and have withdrawn older versions of their Apps from the Apple App Store - YouTune being a prime example. Withdrawal of Apps supporting older models of iPad is making their utility and usefulness significantly more limited, especially so if you ever have to reset your iPad you will have to reload many Apps. Once compatible versions Apps are withdrawn, you’ll not be able (re)download or (re)install them.
If Apps that you need to use now have higher minimum iOS-version requirements, your only option is to replace your iPad with newer model. Of current iPad models, even the entry-level iPad7 supports current iPadOS 13.x - and has hardware specifications and features will significantly outperform your old iPad. Other models of iPad offer a further step-up in both performance and features.
I hope this information is helpful - if only in clearly stating your current options.
