I find the scimitar a mysterious weapon, because curved swords are so unwieldy. The curvature of the blade means that the user has to bend his wrist forwards more to get the blade to bite, which makes the weapon awkward. Also, the curvature means that twisting the weapon to move it in a new plane requires more time and effort. That they may have been better for slashing is easy enough to understand, but in a fight the tiniest fraction of second's delay can mean the difference between parrying in time and being killed, so the wieldiness of a weapon is close to paramount.
One explanation is that they were used in conjunction with a shield, and the shield was used for parrying, and the scimitar was used with dirty-great hacks and slashes. Another is that the user used athletic leaps and bounds as one sees in kung fu films and wushu displays, and by this method kept himself safe. I doubt that this last technique is much use in a big battle.
A third explanation came to me when I met a man at a re-enactment event who told me that his father had been on patrol in Lebanon in the 1970s when three men burst out of a crowd and attacked his platoon. The British squaddies were armed with FN rifles, which had long barrels and a very powerful round that would put a hole in a man, and the man standing behind him. The assailants were armed with scimitars, which they used without shields, and by holding them very close to their bodies, blunt-side-in, they were able to cut and sometimes use short thrusts from very close quarters. Seventeen men of the platoon were wounded before eventually the soldiers were able to step back and get a clean shot that wasn't going to kill bystanders or other soldiers.
How true and well-remembered this tale is I cannot say, but the way the weapon was used struck me as convincing. With this technique, the scimitar could be used without a shield. The technique still suits duels and skirmishes rather than dense mass battles, but I have no reason to believe that scimitars were very unlike swords, in that they were not the main weapon of a soldier, but were instead personal defence weapons. Spears, bows, and halberds were the main battle weapons - the equivalent of the rifle. Swords and scimitars were more like pistols - soldiers liked to carry one when off duty or for surprise close encounters, but no army ever went into battle armed mainly with pistols.