Well, I shot Archery for quite a while, what needs to be uderstood here is that the size of the bow it not actually so important as the tension of the string itself. Also, horse-bows tended to be recurve bows, which curve back on themselves, thus giving extra surface area.
The British Longbowmen who served aboard US navy vessels string their bows at around 110lb. That is, quite likely, more than most people could pull back more than an inch or two, but these guys were trained to pull them full stretch. Longbows had an advantage over pistols of longer range and faster reload, but required far more training and practice. It is
that which killed the bow, not it's effectiveness. A Bow is still more effective Kill-range wise, than many pistols.
Arrows are also often a part of Medieval Bows that are ignored. You don't just get 'pointy' heads on them, you got Armour-piercing points, with Bullet-shaped heads and hardened tips, you got 'rippers', which were vicious inverted blades, designed to be removed only with the maximum of damage.
A decent Archer can hit the Bullseye of a target 60m away without using any kind of telescopic sight, only the sight on the Bow. A good Archer won't even need a few practice shots to set up the sight.

Longbows, such as the sort used by Welsh Longbowmen, could effectively rip an armoured unit to shreds. Only the Crossbow was more effective and that was because of the nature of the Bolt, not the power of the weapon.
Edit : And note I say 'could' not 'would', Archers had their weaknesses. Weather affected them greatly, as did Ammo reserves. The shorter the range, the higher the chances of piercing armour, but the less time to get out the way. Life as an Archer was never dull
