If you are using a current browser, their renderings should be pretty near identical. Plugins such as adblock-plus might remove ads, but they should be pretty close. What browsers are you comparing? More differences will be noted between desktop, tablet and phone. Is your site responsive?

I have problems when viewing Chrome.

FarmBoy

correction - I have problems when viewing with Chrome.

FarmBoy

Turn off all your plugins. Is this a current version of Chrome? Is the site layout different? You see different images? Please be more specific about the problem. Did you delete your cache? What other browsers have you used in testing your site? Firefox, Opera, Safari, IE, Brave, Midori? Platform (desktop, tablet, smartphone)? HTML5 differences [html5test.com]

The only way a browser can make a difference is if old information has been cached, overriding a more recent 301. Even then, you should get the new material if you refresh the page, which is supposed to force the browser to send in a fresh request. “Empty your browser cache” is in any case a good all-purpose fix for what ails you, like “replace your tenor” or “clean the spark plugs”. She said, dating herself.

But just to clarify: What, exactly, do you mean by “forward”? Do you mean redirect, such that a request for
example.backup/dir/pagename
gets a 301 redirect to
example.com/dir/pagename
? Is this happening at the server level, or via a script that executes after the user has already loaded the page?