Safari is an integral element of iOS - and cannot be updated independently of the OS.
Whilst your iPad4 is obsolete this is simply due to technology moving onwards. Sooner or later, computing devices reach the end of their useful life.
The iPad4 was released in November 2012 - and is now well in excess of eight years old. Update support for iPad4 finally 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. The 32-bit platform upon which the iPad4 is based is obsolete.
Many App Developers have also dropped support for older devices and iOS versions preceding iOS11/12/13 - and have withdrawn older versions of their Apps from the Apple App Store. Withdrawal of Apps supporting older models of iPad is making their utility and usefulness significantly more limited. Once compatible versions Apps are withdrawn, you’ll not be able download or install them.
As for the Safari browser, it is an old version that no longer has the capability to support many of the features demanded by modern websites. As you observe, the websites themselves declare that the browser is too old.
If Apps and services that you need to use now have higher minimum iOS-version requirements than your device can support, your only option is to replace your iPad with newer model. Of current models, even the entry-level iPad8 supports current iPadOS 14 - and has hardware specifications and capabilities that vastly outperform your old iPad4.
You also make comment as to the condition of the battery. Given its age, it is perhaps not surprising that this too is approaching the end of its usable life. It is testament to the quality of Apple products that the device continues to work at all. Likewise, for the iPad to continue in mainstream support for five years is industry leading; other manufacturers mobile computing products may only receive product and App updates for only one or two years - some never receive an update.
I hope this information is helpful - if only in clearly stating the limitations of your iPad4 and your available options.