I have NEVER heard this shit outside of America for a very simple reason. In the US, there's that distinction between federal government, and the states. And naturally, having the federal government take more responsibilities means less state influence in matters. Which means for the brainlets that the left wants to take their guns, their house, their cash and their penis. Why ? Because for "conservatives", the states' rights must not be infringed because of tyranny and so on. Yet, for some reason, you always hear this fucking stupidity coming from them when the Federal government tries to create a better healthcare system, and not when there are federal agents gassing protesters. Worst, "small government" proponents wish to destroy federal legislation in all matters, pretending it is best left to states themselves. At best, they're complete dumbfucks who buy into the propaganda that less federal influence/state programs means more freedom, more better, more gooder. At worst, they're complete hypocrites who deliberately wish to weaken the state's ability to intervene to better the lives of the poor, or to protect the environment; so that private/corporate interests may benefit from this.

So yes. Big government intentionally bears negative connotations for the average joe, thanks to the completely unbiased media, who definitely aren't paid to spread bourgeois propaganda. If you dare say that you're for government programs, they'll accuse you of being totalitarian, wanting big brother to take over to turn the USA into a communist country.