Never, ever, underestimate the role of fashion in sword design. Long swords were elite weapons. (“Long” here refers literally to the length, and not to any particular style of weapon.) Elite weapons were desirable because they announced your status merely by wearing them, without even fighting.
And in martial cultures, you spent a lot more time standing around wearing swords than you actually spent in combat using them. So their social utility was at least as important as their martial utility in terms of understanding why they looked the way they did.
Medieval European sword design inherited from Roman antecedents, and in particular the spatha, which was a design inspired by Celtic long swords. As a long sword, the spatha was expensive and well-suited to cavalry, so it established itself as the continental sword of the aristocracy quite easily, and the aristocratic sidearms of Europe were dominated by long, straight, double-edged blades for many centuries thereafter.
Curved swords existed side-by-side with medieval long swords for most of this time, and they were, in fact, very popular, so it is untrue that European swords, in general, were in the straight, crucifix style. But curved swords were not considered “elite” weapons. They were short, cheap, practical weapons like falchions, which drew from a long tradition of agricultural or sickle-style weapons that were associated with peasantry and commoners.
They were looked down on as status symbols, even when regarded as useful weapons. Even those who could afford horses and fancy long swords would often carry a short curved sword like a falchion with them for when things got down and dirty in the melee.
The Muslim world (as encountered by Europeans) overlapped substantially with the ancient domains of the Roman Empire, and its notions of elite swords were not dissimilar. During the Crusades, Arab swords were typically long, straight, double-edged, and single-handed.
They had a bit less of the crucifix-style hilt going on, but that’s mostly just the guard design, and that didn’t come from the Romans, anyway. So there really wasn’t much in the way of “international influence” to steer western sword design away from the straight long sword.
Until the Turks. Although curved “sickle swords” were known throughout the world, their roots in agricultural labor didn’t give them a lot of social cachets. The Turks may have been the first to lengthen the curved sword into an elite cavalry weapon, somewhere around the 8th Century. (There is also a short Turkish curved sword called a yataghan, but it did not have nearly so much influence.)
But even so, the peoples of the Asian steppes were treated as barbarians by most of the cultures and empires who encountered them, so the mere existence of a long, curved sword that was useful from horseback would not have been enough to convince western aristocrats that it was a proper badge of status.
But after the Mongols and then the Turks overran much of western Asia and established their own empires, perceptions about elite weapons began to shift. What does “elite” mean, after all, other than “associated with the ruling class”? After Turks have been running the show for a while, their weapon styles began to redefine the general conception of “aristocratic cool”. (And kicking some highly respected ass all through the region certainly didn’t hurt the reputation of their weaponry, either.)
This happened earliest in lands like Persia and India (giving us weapons like the scimitar and talwar), but it eventually came to Europe as well, with Ottoman incursions into, and rule over, eastern Europe in the 14th through 19th Centuries.
This led to the European take on the Turkish sword, which is generally known as the sabre, and is so solidly ingrained into our own cultural patterns that we hardly recognize its Turkish influences anymore.
That’s partly because the sword was popularized in Western Europe by Hussar regiments, which were modeled after the forces that drove the Turks out of Eastern Europe. Part of its popularity was that it was perceived as the weapon that defeated the Turks, when in fact the opposite was closer to the truth.
So fashionable was the new curved sabre that it became the dominant style of the western military sword during the 19th Century, and although the classic straight-bladed, double-edged sword (by then called a broadsword) did manage to survive, it mostly did so by rebranding itself as a “sabre”. (See, for example, the Patton saber, and the 1908 trooper sabre, both of which are basically rapiers masquerading as broadswords, while calling themselves sabres.)
(This post covers a lot of ground, but Swords and Hilt Weapons is a good overview of the broad historical development of the sword, with Barbarians and Christians and 17th Century Europe, both by Anthony North, covering the main influences on western sword shapes.)