Aye, swords are actually fairly quick and nimble weapons, especially given their relatively light weight and good balancing. Sure, most European blades 'cleaved' more than they 'cut', but they relied more on the momentum behind the swing rather than the sharpness of the blade- though a weightier blade definitely helped with extra momentum. 

Greatswords in particular are actually quite quick. With how long the hilt is, you actually have quite a lot of mobility and maneuverability with it, the upper hand guides and aims your blade, while your lower hand just drives it forward with some good old fashioned lever action. After that, the heavier blade and the length does the rest of the work.

adding on to this- not that fishing was arguing that rapiers are bad, but just fun facts: The reason why rapiers are such good weapons is is because of their length and mobility. Rapiers are light compared to a greatsword or a longsword, but they definitely aren't light weapons- they come in pretty close in weight to a 1 handed arming sword. However, unlike the arming sword (or other cutting weapons like it), that are balanced several inches in front of the hilt, the rapier is balanced very near to (or occasionally even on) the hilt, allowing a user to pretty easily guide the tip of the blade and thrust with it. That comes at the cost of being able to put force behind your cuts, but a rapier while capable of cutting, is really designed specifically for thrusting with its long blade.