Historical tests had the result that a sharpened longbow can cut through a katana.
No really, I am no expert in bows, and I think this topic was derailed heavily, but my personal opinion is that I agree that the longbow probably was the "strongest" bow.
Since I said I have no clue what I am talking about my opinion isn't really of any value, but hear me out anyway:
I guess we agree that for a long shot with a bow you need a lot of energy. The more energy, the longer the shot. That's obvious.
Now if we look at the bows, I think we all agree that the English/Welsh longbows required the highest drawing power of all bows. I mean just look at the backs of the people who used them! And I think it is relatively safe to say that the average Briton was a bit taller and bigger than the average Turk/Arab, I think what is true nowadays wasn't different back then.
I also think it is nice to know with which techniqes the bows were made, be it recurve, composite or whatever. But in the end it is only determining how the bowyers dealt with the materials they had at their disposal. To stick with a similar example: you can lift an object with a wind either directly or by the use of a pulley. One way (with the pulley) feels easier than the other, because you don't have to put that much energy directly into lifting the object, it's rather spent in the "endurance" which you need to turn the wind a few times more than when lifting the object directly. I think it's "maximum power" vs. "endurance power". It's at least how it's said in German, I couldn't find a translation. BUT: once both objects are lifted they both have the same amount of power stored. So in the end I don't think that anything but the final draw weight matters. Every bow offers a certain resistance, and you have to overcome it to put energy into the arrow. The more resistance you have, the more energy can you put into the arrow. And it doesn't matter how that resistance is created, be it the material or the shape or both, in the end it all boils down to how much force you need to pull the string, since for every historic bow it has been done the same way. If there had been a compound bow I already wouldn't be so sure in my theory, since I can't really estimate the shift of power needed, but I think since the drawing length is limited a compound bow would be weaker than a normal bow of the same size, right?
So bottom line is: the Longbow had the higher drawing weight, which means it puts more energy into the arrow.
But here is the point: I think the arrow kicks in, too! My bet is that the arrows for longbows were much longer, thicker and thus heavier than those of the Turk bows. And not only proportionally, but even heavier than that! In Europe I think you had to deal much more with (plate-)armour than in the middle east, and the fighting was different in general. So the heavier arrows lead to the longbow perhaps not having the range of the Turk bow, but his arrows pack a much meaner punch than those of the Turk bow!
I think we can compare it with modern calibers a bit. The Turk bow is a 9mm gun, whereas the longbow is a .45 ACP. While the 9mm has a good range of about 200m or even more, the .45 ACP is so slow, it's subsonic ammunition by default! (In most cases, there is always some special cartridge). With tracer ammunition I could fire my gun and then watch the bullet travel to the target... But heaven forbid you get hit by a .45!
The .45 has a much bigger cartridge than the 9mm, but due to the heavier projectile it's slower and thus has the shorter reach. Still it hits much heavier than the 9mm.
And the same thing happens with the longbow. It was designed to be able to punch through plate armour on short range with the right tip, something which a man with a melee weapon often fails! And of course it was designed to engage the enemy as soon as possible, so they also tried to get the range as high as possible, but it was only secondary to the armour piercing aspect.
The Turk bow on the other hand had the main purpose of having the longest reach possible. In the middle east warfare was different than in Europe, it was more mobile, and they relied much more on ranged fighting than on melee, and in the end a lower percentage of the Saracen troops was armoured compared to European troops, and the armour usually was a bit weaker, as far as I know. (I know about the fact that they often wore cloth over their armour so the Europeans thought their majority was unarmoured, which is wrong). So the main objective was to be able to shoot before the enemy could shoot, and be it only for hitting their usually unarmoured horses first, depriving them of their mobility.
But since I think that in a direct fight of lightly armoured Muslims against heavily armoured Europeans the longer reach of the Turk bow would be negated by the lower stopping power, whereas the longbow has only slightly lower reach but is more deadly, I think the longbow is the more useful tool in general. Edit: but only because I had to consider the armours of the possible enemies, too.
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