Safari 9.1 was launched in 2016, more than 3 years ago, and does not support a much of popular web technologies.
Polyfills exists for some technologies, with Safari 9 just making the list for some polyfills, but many developers do not expect it to be compatible with a browser that lacks ES2015 support (you can not even declare a variable with the Safari 9 option).
It is not possible to know exactly what is causing a problem without having a better idea of what technologies are trying to access the specific websites with which you have problems, but caniuse.com offers a detailed comparison based on features of web browsers, including Safari 9 vs Safari 12.1.
If the web developer has not included a polyfill for Safari 9, it may be possible to inject it yourself for individual web technology functions, but this situation will only get worse. I understand the potential need to run older software, but this use should be limited to a virtual machine, ensuring that the host operating system is secure and running a secure web browser capable of understanding the technologies present in your favorite sites.