There has been a lot of progress recently on the web-platform-tests project [1]. This is a cross-vendor effort to improve interoperability on the web platform through testing.
Historically the process for building web-platform features has been to write a spec and then assume that implementations of the spec would reach compatibility in an ad-hoc manner by patching until sites worked. The big innovation of the last decade — still controversial to some — was the idea that it's OK to adjust the specs themselves when implementations have converged on some other behaviour or when the spec is otherwise wrong. The goal for the future is to use the same engineering discipline that you would use for developing software to developing the platform itself. In particular the objective is:
* Every change to a spec that influences browser behaviour must be accompanied by a corresponding test case.
* Every change to a browser that affects a cross-platform feature that isn't already adequetely tested must be accompanied by cross-browser test cases.
* The results of those tests must be visible to browser vendors in a way that makes it easy to identify the higest value bugfixes (e.g. cases where N-1 browsers agree and one is different, or cases where a spec change makes previously correct behaviour — which may not have shipped yet, but just be enabled on nightly builds or behing a flag — wrong).
Apart from the work of actually writing the test, achieving those goals involves a lot of infrastrucutre work to ensure that the cross-browser tests are well integrated into the development cycle of each browser and are able to cover as many scenarios as possible (testing often involves manipulating the browser in a way that is not exposed to normal web content). Despite the large scope of the project, I think it's agreed that modernising the way the platform is developed, and prioritising interoperability at all stages of spec and browser development is essential to avoiding existential threats to the open web in the long term. Certainly Mozilla put a lot of resources into post-hoc fixes for site-compatibility issues, and whilst there will probably always be some bugs that slip through, it will be much more efficient to catch those problems up-front before they ship to end users.
I sadly don't have data to backc this up, but I'm pretty sure we're already seeing the effects of this effort, with recent, complex, features shipping with fewer cross-browser issues that we would have predicted five years ago.
[1] http://web-platform-tests.org/