I don't know that I agree with your premise.
There are only two swords in Game of Thrones that are intentionally depicted as heavy that I can think of off the top of my head. The first being Ice (Ned Stark's old sword), which was largely a ceremonial sword and used for executions, not really a war sword. The other is the Mountain's sword. But the Mountain is supposed to be using a massive sword because, well... he's the Mountain.
All the other sword fighters in the show, Brienne, Jaime, Syrio Forel, Bronn, The Hound, Arya, Jon Snow etc all seem to handle their swords pretty deftly.
Sometimes if the actors are not that skilled with technique something may seem awkward and therefore might be interpreted as the sword is heavy, but I think they do a decent job. They're not using Liechtenauer and Fiore to a tee, but I've seen much worse swordplay on screen.