Safari 9.1 was released in 2016 over 3 years ago and does not support a party of popular web technologies that have since been developed.
Polyphilips exist for some technologies, with Safari 9 barely making the list for any polyfills, but many developers do not expect to support a browser lacking ES2015 1 support (you can't even declare a variable to leave Safari 9).
It is not possible to know exactly what causes a problem for you without more insight into which technologies are trying to access by the specific sites you are experiencing, but caniuse.com provides a detailed feature-based comparison of browsers. including Safari 9 against Safari 12.1
If the web developer has not included a poly 9 for Safari 9, it may be possible to inject polyfill itself for individual web technology features, but this situation will only get worse. I understand the potential need to run older software, but such use should be limited to a virtual machine, so the host operating system is safe, and running a secure browser capable of understanding the technologies found on your favorite networks.