Personally, I suggest that the best solution to your problem is to use websocket. That's the most effective way so far. And regarding this:
They don't allow webrtc or node.js or anything like that.
You can find some hosting that can allow you to run websocket servers. Like for example https://www.digitalocean.com where you can setup/install your own socket server in the hosting.
I have a messaging system using PHP/SQL/Ajax (When user's viewing the message thread, it fetches new data every 60 seconds) but that doesn't seem instant and also seems like it's not very convenient for the server if a lot of users are on.
I think this method you currently use right now is AJAX Polling where you persistently request on every interval for new updates in the database. This is okay for minimal updates but personally I don't recommend using this method. At every request per interval is affecting your app speed and performance in the long run. That's why you are correct:
I'd like to implement this with notifications and comments as well, so having a lot of ajax scripts requesting data every x amount of seconds doesn't seem like a good idea.
And this:
So, my question is: Is there any way I can live-update data to my users without having to constantly request new data and not having to use websockets?
Yes, there a way using SSE (Server-Sent Events). Check an example from w3fools :D.
But just in case you consider websockets again, you may try websocket for PHP: Ratchet.
Also check this out: Ways to make request/get realtime updates for a web application.
