Game of Thrones is a medieval fantasy which still features women, blacks, gays, and other minorities in positions of power and strength. However, Fat People are universally picked on. This seems to reflect on our society more than anything resembling a medieval truth.
I study and write about medieval culture. In the actual medieval culture, the class distinctions were much stronger. The prejudice against blacks was stronger. Gays would have faced serious issues. Intriguingly, in a time of war, such as this one, women would have had far more rights. When the crusades were going on, many towns were run by women, and nearly all brewers were women. It was considered normal and natural.
When it comes to "heavy people" that was a sign of wealth! Of health! It was a state to aspire to. It would be like a woman or man in modern times ogling Angelina Jolie and thinking that was an ideal body shape. It was a sign of having it all.
Clearly what we get in Game of Thrones (and really, in any modern movie / tv series) is a reflection of our modern values. We want it to have the "flavor" of the time period in question, but the way people look, act, and think reflect our own current culture. So we have a great deal of homosexuality in the series. We have a lot of sex. We have women talking about power and striving to have it. We have blacks shown in positions of strength. Daxos is a powerful man in Qarth and it's not even blinked at that he's black. There are never (that I recall) any denigrating remarks made to people simply because they're black. We modern people would consider that "inappropriate".
But people are ROUTINELY harassed for being fat.
Sometimes it's insidiously minor. When Robert Baratheon's squire is trying to put Robert's armor on, and failing, Robert snaps, "Your mother was a dumb whore with a fat ass". So being fat is right alongside being stupid and a prostitute.
When Jamie is first talking with Alton, to lure him in, Alton says his mother was Cinda. Jamie asks, "is she the fat one?" When Alton stumbles over an answer, Jamie steps in and says with derision "there's only one fat Lannister" and that Alton would know if it was his mother.
One of the only actual characters shown who is fat is Samwell Tarly, one of the Night Watch. Now, first, let's discuss what current culture thinks of fat people. Study after study shows that the perception of a fat person is that they're slow, stupid, smelly, and incompetent. Studies even show that a person sitting NEXT to a fat person in a waiting room for an interview is then "tainted" by that contact and is rated lower on all those scales. So how does Samwell's character hold up?
Yup, you guessed it. He's immensely cowardly. He's slow. Fearful. Afraid of his own shadow. Yes, he does some reading, but it's not portrayed as a positive thing. His fellow watch harass him incessantly. When they hide out with Craster, Craster openly tells the others to eat Sam because he's as fat as a pig. He tells them to carve off pieces of meat slowly.
Then we have Hot Pie, the boy who runs away from Harrenhal with Arya and Gendry. Once again, Hot Pie is shown to be cowardly, slow, and grumpy. When they're found by the Brotherhood, Hot Pie is simply called the "fat one".
Do we have anybody called unhealthy for being too skinny? No. Skinny is in in our modern culture, and it's reflected in the dialogue. Women are praised for being slender, rather than richly curvy.
King Robert Bartheon himself is fat, and certainly Stark makes a comment or two about it. Again, while this would have been a sign of wealth in medieval times, here it is looked down as a sign of being degenerate, along with his heavy drinking and whoring.
It's intriguing that we are in a modern world where talking openly about homosexuality is praised, and accepting openly that blacks and other minorities "of course" can hold positions of power is normal - but at the same time routinely and regularly harassing fat people is simply the way things are. People don't bat an eye, and they're even supposed to laugh.
Hopefully some day we will be at the point where this behavior against fat people is considered in the same light that behaving that way against blacks or homosexuals would be.