So there are historical sources that claim that a Mongol composite bow have effective ranges of over 400 yards while English longbows 300 yards. As someone who has competition archery experience at the college recreational level (I’m in the archery club) and who follows world class archers, I can tell you that these values are hyperbole. Perhaps these are the maximum ranges that medieval archers can shoot in a parabolic arc, but they certainly will have no accuracy whatsoever.
Here is an Olympic bow https://m.media-amazon.com/images/I/31x9EaUx88L.\_AC\_UF1000,1000\_QL80\_.jpg
It has a boatload of accessories that enhance its accuracy, from stabilizers to sights to ideal arrow rests. The purpose of Olympic archer is to take traditional archery to perfection, as in being as accurate as possible with recurve bows. Yet in official competitions, the distance to hit yellow is 70 meters. To snipe a human sized target, they might manage the same accuracy at 140 meters which is far less than the fabled 300 meter range of a longbow. And while there might be prodigies in the army, the AVERAGE English longbowman is nowhere near as accurate as an Olympic archer. That’s before considering that Olympic archers shoot 50-60 lb bows while English longbowmen shot bows of 110-150 lbs which further reduces accuracy. And then Turkish or Mongol composite bows with 400 yard + effective range? Travesty.
What about barebow competitions? If Olympic archers are only accurate to so far, then barebow archers are even less accurate. \[Marco Pontremolesi, gold medalist longbowman\] is a top barebow shooter accurate enough to hit yellow at 55 yards. So he can snipe an enemy soldier from 100 yards at best. That \[2019 Indoor Archery World Series event\] was the best performance I could find for barebow archery. The competitors had abysmal accuracy. And again, these bows were lighter in draw weight than medieval warbows.
You may also believe that medieval warbows had longer maximum ranges than modern bows due to higher draw weight. That is simply not the case when you look at modern recurves and compound bows. https://www.usarchery.org/resources/2021-flight-records-261021142708.pdf
So compound bows have 3x the range of English longbows, and even recurves with a fraction of the draw weight shoot farther.
From this information, my theory is that bows used in warfare were shot en masse, against an enemy army. They were not precision weapons, certainly not in the claimed distances of historical sources. There is no reason for precise shots when there’s an entire enemy column advancing towards you, so archers shot in volleys rather than individually.
TL:DR The effective ranges of medieval bows are the distances at which they were shot in volleys, not accurate shots that sniped enemy soldiers.