There are plenty of synonyms for the phrase "run ahead" that can be used in different contexts depending on the situation. For instance, in sports, it might be replaced with the terms "dash," "sprint," or "race ahead," all of which connote a sense of speed and urgency. Other synonyms for "run ahead" include "lead the charge," " forge ahead," "boldly push forward," and "pioneer the way." Meanwhile, if you want to describe someone who is being too forward or presumptuous, you could use "jump the gun," "pre-empt," or "push the envelope. Whatever the situation may be, there is always a synonym to use instead of the words "run ahead."".