Being able to fire multiple arrows in short succession is actually a trope, usually because the increased firepower is desirable from an action standpoint, but the limitations of the time period will not allow firearms.
Your main problem is that you're trying to fling multiple projectiles with a single bowstring. I'm no physics guru, but it seems that you're going to be dividing the pounds of pull across the total number of projectiles -- so you'll get correspondingly less range & target penetration than if you stuck with a single arrow (ignoring more obvious issues like aiming). Someone actually ran the numbers on how much kinetic energy from the bowstring is transferred to the arrow, if you're interested.
A crossbow is a much more workable solution, as the strings can be drawn in advance, and held in place. That leads to the possibility of firing more than one projectile, in quick succession.
One solution to firing multiple arrows is a special crossbow, featured in a battle scene in the 2000 film The Gladiator. On first glance, it looks rather ridiculous, but I actually feel it is a workable design (albeit perhaps not terribly practical for anything other than a showy arena skirmish) for the simple reason that each crossbow bolt has its own string, rather than some sort of imaginary bolt "magazine" that ignores the necessity of re-cocking the string after each firing. Additionally, it dispenses with any sort of fanciful common trigger system, and appears to have the wielder just rotate the entire weapon on its axis.
A simpler version would be the double-sided crossbow, from the 2000 film Dracula:
