how to fix web applications bugs without breaking other parts

Assume you have a WordPress theme and you are trying to fix a few bugs in it. However sometimes things go wrong and other parts of your application can get affected and new bugs are created. So ultimately this can go for ever...

I know there is not a clear answer for this but I would like to hear some best practices. Is it mostly because of design pattern or testing procedures. What can we do to prevent this without refactoring the codebase?

Thanks.