Note that the bolt Payne-Gallwey shot 440ish yards weighed 3 ounces (85g). No hand-drawn bow can send an arrow of that weight so far. While the Mary Rose arrows appear to have been in the 45-65g range and another surviving arrow is around 50g, Matthew Strickland and Robert Hardy in The Great Warbow argue that heavier arrows - up to 4 oz - were common. Various extant Manchu war arrows are 100g and over, so this isn't hard for me to believe.

As far as accuracy, in addition to Humphrey Barwick's claim, I'll mention that Fourquevaux gave a specific account of a single crossbower a siege in the 1520s who killed more of the enemy than five or six of the best gunners. Unlike Barwick, Fourquevaux considered both bows and crossbow more accurate - or at least more reliable - than guns.