How to balance "ship now, fail fast" attitudes toward product development with making sure tests aren't sabotaged due to insufficient QA/sloppy design?

I have a client who does radical redesigns of sales pages/funnels: They're gung-ho about launching and running a test as fast as possible, regardless of issues raised/bugs discovered.

If you have advice/cautionary tales/thoughtful articles to share about the importance of comprehensive QA before running tests, especially ones with radical changes, I would love to hear them!

Thanks in advance :)