Continuity mistake: Young Forrest is about to step in the bus. He says to the bus driver, "My name's Forrest, Forrest Gump." That moment we have a back view of the bus driver's head and she exhales a big puff of smoke. A split second after we cut back to a front view of her and there's no smoke, instead she immediately exhales the smoke again.
Continuity mistake: Before young Forrest's foot gets trapped in the sewer, a black car in the back is moving towards him, and is about to make a right turn. Close-up on Forrest's foot and, when back to the wide angle, the same black car is further back again and repeats the previous movements.
Continuity mistake: When Forrest was running in the first football scene, it shows cheerleaders and a couple of football players on the sidelines, then the next scene in closer view of Forrest running it shows the same cheerleaders and players in the same exact spot as if Forrest ran that same part of the field again. (00:22:15)
Continuity mistake: When Forrest gets hit by a stone and falls on the ground a red school book is lying on the ground next to him. When he gets up he doesn't pick it up, and when he runs away it's not on the ground either. (00:15:20)