No, it's not the government censoring people necessarily, but it still goes on, most notably in the media. A good example is many media outlets refusing to show or print Charlie Hebdo's first cover after the attacks. It could have been an opportunity to take all of the power away from these extremists by painting a target on everyone's back, but they chickened out instead of showing solidarity. Another example is some liberal politicians refusing to use Islam in the same sentence as terrorist and even going so far to make ridiculous statements like it being a "religion of peace" when it really isn't (none of the Abrahamic religions are, at best they're patently hypocritical when it comes to violence). Yet another example is Sanders and other liberals continuously bringing up the wage gape when it comes to men and women and often misrepresenting what it actually means. You can't force a group of people to seek out higher-paying jobs if they don't want it in some odd quest for "equality". It's just disappointing as someone left-leaning to agree with a lot of what the left stands for, but pussying out on some issues because of this intense fear of offending or alienating people.
And of course, this goes beyond just the political campaign. I'm not going to make excuses for the idiots on the right that use their opposition to some vague sense of PCness as a rallying cry, because quite often it means nothing, especially where Trump is concerned. However, it does exist, and not only on the left. It sort of exists for conservatives as well, especially when it comes to criticizing the country, the military, the police, and the leaders that they consider sacrosanct (who aren't liberal or Obama of course). They of course don't call it political correctness though since they're the main ones responsible for turning it into a pejorative.