I'm a Brit and had a similar moment of realisation when I discovered America wasn't about the size of Spain + France. I think it was the facial expression my (American) girlfriend gave me when I suggested we drive for a weekend away in Florida (we were in New Jersey at the time) that tipped me off.
America is basically the size of Europe, which really surprised me. But then, it's not knowledge you ever really need to know.
No, America (meaning the USA) is somewhat smaller than Europe, especially if you count European Russia, another country whose size is often underestimated. Though, it is helpful to realize that the United States is, in fact, larger in terms of landmass than China, and is exceeded only by Canada (the other North American Anglo democracy, much of whose land is largely empty due to the extreme cold) and by Russia, (another country with a lot cold, mostly unused land.)
Russia, even in its post Soviet scaled-back state, is in fact largely an empire. Most of its land was conquered from indigenous peoples who still inhabit it, this it not too different from both the United States and Canada, though the circumstances are somewhat different.
It's smaller, yes, but not by much. The USA is 9.8 million km^2 and Europe is 10.2 million km^2. Though that does include Alaska.
I think the thing that took me the longest to "get" was simply the distances between things. If you'd asked me a few years ago how far one coast of the USA is from the other, I'd probably guess at around a thousand miles. Top to bottom maybe 500.
Mind you I also didn't realise Washington wasn't in Washington state.
My parents and I live in Texas, and they went to New York a few years ago on vacation, renting a car and just driving around for awhile.
They'd take out a map, determine their approximate route from point A to point B, figure it would take about three hours... and arrive, confused, 45 minutes later.
Until I started dating an American girl, I can't say I was ever in a situation where I needed to know how big America was. The US just isn't very significant to the average European.
It's interesting that you selected Spain + France. Texas + California (our 2nd and 3rd biggest states) are only slightly smaller than Spain + France. Alaska (our largest) is bigger than France, Ukraine, and Sweden combined.
America is basically the size of Europe, which really surprised me. But then, it's not knowledge you ever really need to know.