Curved swords, especially very curved swords such as the Turkish shamshir (or scimitar), slice. The blade is drawn across the area struck like the action of a bacon slicer. Wounds from such a blow can be devastating - loss of limb - decapitation etc. The curve of the blade also allows blows to be made in crowded situations (as someone else has said). The major draw back of this type of sword is that it makes the thrust essentially impossible - I know I've tried, it's like trying to thrust with a three-foot long banana!

When used as a cutting weapon a straight - double edged sword hacks like an axe into wood, this can also give devastating wounds, but only if the sword is heavy-bladed, in turn this weight cuts down on its maneuverabilty. The big advantage of the straight double edged sword is that it can be used both to cut and thrust. The thrust is the surest way to kill quickly, if delivered correctly. However, a thrust, if it doesn't hit the mark, results in you being over extended and wide open to your opponents riposte cut.

In war the curved sword would have the advantage in melee conditions, though the straight sword would be more useful in the first clash, when opponents are at arms length. This is particularly true of cavalry charges where a straight sword thrust can make use of a locked wrist to allow the impetus of the horse's motion to be directly used.

Further, if the enemy has well-made, effective armour the utility of the curved sword is particularly reduced as the armour-piercing ability of the thrust is much more effective than a cut from even the sharpest sword.

All this said the cut is the more instinctive way of using the sword, and the effective use of the thrust requires more skill and training. Perhaps the bad guys were less skilled warriors, in general, than the good guys?