There are a few things that might be happening that break your site.
It can be that indeed you are using a plugin that has compatibility problems.
Or another case you need to keep in mind is that some of your extensions could contain customizations on their core files, that get overwritten with the update. So any code that was responsible for certain functionality or layout you currently see in your website are removed and you lose this customization.
Or it can be due to an overall bad setup/implementation of the site from the beginning, using a combination of bad practices (using already old extensions that are not supported anymore, bad custom coding, bad template implementation, too many extensions that conflict with each other, etc).
Compatibility issues usually will manifest with php errors - so to see if this is the case, enable error reporting and look for errors printing in your screen. Sometimes it might be just a simple fix, to update a part of the extension's code to be compatible, some other types might be more complicated of course.
Now, you have a valid question when asking how to know an extension is compatible, before you updating.
There is not a straightforward way to know this through the Joomla Backend. I would recommend you to become familiar with the set of 3rd party extensions you have installed on your website, so at first phase you know which ones you are using, where, why and how you are using them.
Then get rid of those that you don't use/need.
For doing so, go in Extensions Manager, and start filtering by extension type to see them all for all the types. Get to know your 3rd party extensions list.
Search also in Joomla Extensions Directory for compatibility and recent reviews, but also in Google, for those extensions and visit the developers websites to find more information. If for example you find an extension that is unpublished in JED, and also you find an outdated and abandoned website of the extension's developer, then likely this extension is not maintained anymore and won't have recent updates nor you should expect any - in this case it will be wise to remove or replace this extension.
Sometimes, helpful information for the extensions like the developers website, or support forums/emails, changelogs, etc... you can find also when you visit them in their backend - or if you look at their code php files.
I just discovered this Google Doc, from a comment by Neil Robertson on another question in JSE, that is listing Joomla extensions that are compatible with J3.5 and php 7. It can be helpful in some cases: