Recurves are generally better. Obviously it depends on the poundage of both bows but a recurve will outperform a longbow of the same draw weight.

First as far as shooting distance goes.. The farthest distance shot with a Turkish bow was over 800m. The record with an English longbow was less than half that at 340m. Modern flight bows also 'happen' to look like Turkish bows (with a few extra gizmos on them.) There's an anecdote of a Turkish ambassador to England visiting an archery range in the 19th century who amazed everyone with a 440m shot, over a 100m farther than anything any of the English had seen. He then apologized for the poor showing, explaining that he was out of practice and his bow wasn't in good condition.

Shooting distance is mostly about speed, so let's look at some speed data. First of all, the arrow speed depends on the energy released by the bow (function of the bow's draw weight) and the mass of the arrow. Because of this, when comparing bows (and arrows) we use a metric called GPP, (Grains of arrow mass Per Pound of draw weight). When you want to compare two bows' arrow speeds, you need to compare at the same GPP. When you collect a bunch of FPS vs GPP data, they look like this.. At the same GPP, Turkish bows have a ~7% advantage in arrow speed. Korean bows have a ~16% advantage (this is because they're shot with longer draw lengths than Turkish style).

The data above shows an advantage in recurves when shot at the same arrow mass, but if you really care about long distance shooting you'll shoot lighter arrows and recurves are much better at that. The minimum GPP you can shoot with a bow depends on how efficient and light the limbs are. If your bow's limbs are bulky and heavy, shooting the bow too fast (because the arrow is so light) will cause the limbs to take damage. You can only shoot a longbow with arrows of around 7GPP and above, most manufacturers void your warranty if you shoot at below 10. Meanwhile because recurve limbs are light, you don't really have to worry about it. Turkish bows can be shot with ridiculously light 1.6 GPP arrows and both Turks and Koreans used arrow guides to shoot very light darts out of their bows.

So recurves shoot better when in the same conditions but also have the option of shooting lighter arrows than longbows are capable of, to shoot even faster. As for why this is, I'm going to copy paste something I wrote previously on the topic:

Yes. I mean if by better you mean better ballistic performance, yes composite bows shoot faster, farther and harder because:

1. Composite bows have more reflex. That is, a longbow when unstrung is straight, a reflexed bow is bent forward in the opposite way. When a longbow is strung there's almost no tension in the string while a reflexed bow has already been bent and tensed quite a bit just to string it. In terms of shooting, this means longbows only accelerate arrows at the beginning of the shot and not so much near the end as the bow uncoils while a reflex packs a lot of energy throughout the bow.
2. Recurved limb tips improve smoothness at the end of the draw. At the end of your draw when the angle between the string and limbtip is significant, recurved tips maintain a more efficient lever point for the string keeps the rate at which the bow gains weight lower. This again means you're packing more energy just before your final draw length.
3. Horn/sinew are more flexible than longbows and means composite bows can be strained more by being pulled more. A higher draw length affects the energy storage of the bow as much as the draw weight does. While a longbow is typically shot at draw lengths of 28" (and expert archers in the past would draw to 30" next to the head), Asiatic draw lengths are typically much higher, about 32", past the head. Longbows can be made to draw longer but that requires making the limbs longer too (see next point).
4. Smaller and lighter limbs keeps the virtual mass of the bow low. When you shoot an arrow, you're accelerating the arrow's mass but the limbs have to get moving in order to pull the string which launches the arrow. This inertia of the bow is called its virtual mass. The heavier and bigger the limbs, the heavier its virtual mass, and less efficient the bow. More energy goes into getting the limbs moving and less energy goes into the arrow.

Points 1, 2, and 3 mean composite bows store more energy, point 4 means composite bows use that energy more efficiently. This is true at all load factors, composite bows will generally always out shoot a longbow (given the same draw weight), even with heavier arrows though people generally assume longbows are better at heavy arrow shooting. It's just that their low virtual mass means composite bows are exceptionally good at shooting light arrows fast and far.

What the longbow does have going for it is that they're easy to make, a good bowyer could turn one out in just a couple of hours and the only maintenance it would need for its lifetime would be some oiling to protect the wood from moisture.

Meanwhile a composite bow takes months to make, sometimes as long as a year depending on the climate because the components have to be glued and let dry completely before adjustments to tillering can be made. Even after it's finished the the bow has to be shot and heat tillered for about a year, before the bow starts to shoot consistently without the limbs warping and flipping front to back. If the climate is humid, the bow will have to be kept in a special heated box to be able to keep the sinew dry for shooting. You pay for the extra performance with a much higher initial cost and require constant caring for.

Both have similar range and "stopping power", the difference is circumstancial:

Longbows are simple bows ie: made from one piece of wood, so they are cheaper to mass produce and maintain/repair, and are more reliable.

Recurve bows are compound bows ie: made from 2 or more pieces that are glued together (usually a combination of soft and hard woods). This gives an advantage in size, so the bow is less cumbersome, and more importantly can be used from horse/camel back, and can be easily holstered when switching to/from melee.

The downside being the glue would sometimes fail in harsh conditions (humidity, low temps, etc...) so it was less reliable in extreme weather.

Bonus for the English archers here would really be the bodkin arrows, they really maximized the ROI of the longbows range

They can honestly be about the same in regards to penetration and power, the difference is in what you want to do with the weapon. Longbows are essentially a big piece of wood bent with a string, and can put a massive amount of force behind the arrow, over 100 pounds in some cases. The drawback is that you're going to have a big bow, over 6 feet for a bow that powerful. So you're largely stationary and not mounted or very mobile, but once you're set up you can punish your foes mercilessly.

The recurve can achieve the same distance and power with a smaller frame, it's more complicated to construct (as you pointed out) , but it can be made with a wider range of materials. So overall it's far more versatile in it's use, but basically can have the same amount of power.

In the end it just comes down to how big a bow do you want to get the job done, both can be just as powerful, and both are going to be just as hard to use when you get into the higher pull strengths.

Edit: In both cases though, most archers in military uses in both regions mentioned were highly trained and very effective, they were professional archers.