That's exactly what they do. It just takes a while since there's tens of thousands of feet of drillstring.
The actual way to avoid the problem is by using bits whose design lifetime in the formation you're drilling is at least as long as the hole section you're planning to drill.
Which suggests just leaving the old drill bit somewhere in the ground. But currently doing that would also involve removing the pipe, as to why: https://news.ycombinator.com/item?id=43371776