What we call search was dominated by Altavista before Google. It really didn't take much to break that dominance.
Stable dominance is characterized by things like network effects or similar. Google, despite being willing to use anticompetitive tactics, hasn't really successfully built much of a moat.
Bing didn't do it since at the time, everyone liked Google and hated Microsoft, and Google was an epsilon better.
Today, Google doesn't have that do-no-evil reputation, and it's hard to know what would happen.