Not to go Trek v Wars, but the starship Enterprise has always had the manueverability of the Falcon despite a mass closer to an Imperial capital ship.
As someone who *does* go Trek v. Wars all the time, you are more correct than you know.
While zipping starships may look silly to our eyes, the simple fact is that the fault lies not in our starships, but in ourselves. A few hundred years ago, tall ships traveling without care of wind direction, and indeed without tall sail of any kind, would've looked pretty silly, too.
That said, for aesthetic and some technical reasons, ships aren't often shown going absolutely nuts, despite the fact that it would be very effective at 300,000km versus lightspeed weaponry to suddenly not be where the beam was fired at a full second ago. (As an aside, there was an old sci-fi story where two warring sides didn't have an arms race in the classic sense. Their battles were won or lost based on the accuracy of their computerized "predictors" that would tell them where to shoot in advance of their foe's arrival to that spot, like a higher tech version of our old automatically-leading gunsights.)
That said, we have seen traces of the maneuverability we would expect. Here I do not refer to the sharp high-speed turn of the Ambassador and Nebula Class ships in the opening scenes of the DS9 pilot (video linked from here: http://st-v-sw.net/STSWcompare.html#STL ), though that's not bad. Instead, look to the graphic of the Kumeh Maneuver from "Peak Performance"(TNG2) in which the Enterprise-D is shown to whip around from behind a planet and then stop cold.
See an old capture here: http://www.st-v-sw.net/videos/peakperf.mpg
I also have some straight-line acceleration demos available here: https://www.youtube.com/playlist?list=PLTua45BtkvDD24H3p1atvl_o5shdwFGXf
Put simply, any ship that can zip off at 1000g should be perfectly capable of having a veritable maneuverability conniption fit on-camera. It may not scale well to our experience, but that's our issue.