Ask HN: How do you test for browser compatibility?

I've taken over for a failing project at work, supporting the largest contract my company has ever executed. We're understaffed in my department, and by taking over, I mean I'm solely responsible for everything. Until now, it's been a jumble of band-aids and temporary fixes. After a couple of weeks of database tuning and other architectural changes, I'm starting to focus on the browser. Unfortunately, the sales people here wrote nothing about browser support in the contract, so we're on the hook for supporting anything the client uses. IE7 tends to be particularly problematic with subsequent builds. Any radical changes I make seems to break a lot of functionality for the client after I've pushed the changes to production, but the real problem is that I can't see it in my (manual) testing procedures before the client does -- I'm just using IE11 with the developer tools mimicking IE7 functionality. Everything's beautiful in Chrome, Firefox, and IE9+ isn't bad at all...

tl;dr: How do I really see if things are breaking across browsers and browser versions? How can I automate my browser testing procedures?