There are few options to do this.
Can you, please, try setting your caching profiles or disable caching at all by going to Administration->Settings->Advanced->Pages and inform me if the issues you are experiencing still persist? Also caching is disabled at Administration -> Settings -> Advanced -> System -> Output Cache Settings and disabled EnableOutputCache and EnableClientCache.
Following the above steps you should make an edit in your theme css and it should appear when you edit a page and not on the frontend. As caching is disabled and the browser still have some previous css loaded it will not display the changes republishing the page and making further changes to css displays them as they are made as now the right css is cached in browser.
If you are using hardcoded links to css files (for example in master pages) they might not get properly cached too so <link rel="stylesheet" type="text/css" href="/css/test1.css?t=633543081724531250" /> you can add a fake parameter after the css file name.
The problem is not directly connected with Sitefinity because no caching usually results in such problems in most browsers.
Also IIS reset will apply the changes in your css. This
might also be useful.
You can force the styles to be allied by putting this code in the code behind file of a web form:
var pages = Telerik.Sitefinity.App.WorkWith().Pages().Get();
Additionally making a simple change in the css should get applied when page is republished if you have the above settings in Administration ->Settings.
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