No full screen after clicking on a link

You have perhaps managed to enable a Safari Slide Over view - this being a major multitasking feature of iPadOS, fully detailed in the iPad User Guide for iPadOS. 

Here is the easy way to change the Slide Over view, to full screen... 

Identify the small grey drag-bar at top-centre of the Slide Over window. Touch-and-drag the drag-bar to the top-centre of the screen - and lift your finger; the Slide Over window will transform into a full screen view.

You may find these overview Apple video guides to be helpful:

After viewing, you’ll be ready for the additional Multitasking Guide:

Use Multitasking on your iPad - Apple Support

The iPad User Guide for iPadOS is also a rich source of information. Here are links to both the downloadable and online versions of the Guide:

‎iPad User Guide on Apple Books

iPad User Guide - Apple Support

I hope this explanation and additional information is helpful in resolving any difficulties you may be having with Multitasking features of your iPad.