As a British man currently living in America, with my American wife, I can say with a fair bit of certainty that most Americans aren't any more stupid than most of the Europeans I grew up around. Saying that though, just because most of the population seems to be perfectly normal in terms of intelligence, culturally and politically America as a whole seems to prize stupidity, which is propagated through the media, and thus distorts the world's view on American ignorance.
This can noticeably be seen is various areas:
Politics - the two party fallacy that is "democracy" in this country is ridiculously bipolar. On one side you have die hard conservatives, with a terrible reputation for completely ignoring tried and tested scientific fact, claiming that their 300yr old Constitution and 2000yr old Bible are the only important documents that should rule how we lead our lives, whilst on the other you have a side so liberal that no one could possibly agree with the majority of their policies. Add this to the fact that the way your candidates for all sides conducting themselves worse than a group of children at preschool, and you begin to see why the rest of the world raises serious questions about the intelligence of a country that wants any one of these frontrunners to become your leader. I know most americans don't want any of these people, but unfortunately your system is the one that has made these people into your "best" candidates.
Ignorance of the outside world - it's no stereotypical comment to say that America knows next to nothing about countries that are not America. I'm regularly told that, as an Englishman, I'm exactly the same as a Welshman or Scotsman. This is not only highly offensive (and is the exact same thing as someone calling a US citizen a Canadian) but it shows off your ignorance of other countries. That was just a common example that I deal with on a daily basis, but the ignorance spreads into virtually all aspects of all countries, and I'm continually astonished by how little the majority of Americans know about other cultures and countries.
Learn that criticism isn't an insult - this is the main aspect of Americans that give them such a bad name abroad. This notion that "We're #1!" and that everyone's a winner is taught practically from the cradle. This is a damaging attitude that Americans own, and consequently leaves many blind to their flaws. That's not to say that America isn't great in certain areas - they excel at many things. However, no country is perfect, and we can all learn from each other. Instead of going on the attack when someone points out a flaw in what you're doing or saying, assuming that they're just out to discredit you, understand that in most cases they are just offering genuine suggestions on how things could improve. This is how we teach each other.
Foreign policy - nothing offends a country/culture that has been in existence for 1000yrs more than a country that has been around the international stage for 100yrs telling them that they're doing things wrong, especially when the alternative is a system that produces the likes of George W Bush and Donald Trump as the favoured candidates. Let people get on with their own business, and realise that they don't like you telling them what to do anymore than you like them telling you what to do. The days of America being the World's Police Force are long over - America has been too discredited over the years to be able to maintain that stance anymore.
Like I said in my beginning statement - my personal experience is that Americans are no more stupid than people from any other countries. It's purely the image that your politicians and media give to the rest of the world that continues to drive this stereotype, and until that changes, unfortunately Americans will continue to be looked down upon by every single other developed nation in the world