Many cities in Europe have regulations as to where skyscrapers can be built and how high they can be built. Until the 80s, no skyscraper in London could be bigger than St Paul's Cathedral, and there are still strict rules as to where skyscrapers can be placed in order not to obstruct views of certain London landmarks from certain vantage points around London (sadly though, the authorities have been known to be lax about those rules provided they've been paid enough...). Paris has even stricter rules against building skyscrapers in the centre of Paris. Most US cities on the other hand are much newer than London or Paris and have few heritage sites or historic panoramas to be protected.

As my predecessor mentioned European countries have always had a lot more restrictions on the building of skyscrapers. Mostly a building can't be much higher than a famous building in a city. At other times it was dependent on the width of the street. I believe this was the case for Paris.

In the US there were also some regulations like the important 1916 zoning resolution. These were more focussed on the buildings getting a more pyramidal shape in order to maintain some sunlight on the streets.

That is not to say that Europe doesn't have sky scrapers today. There are plenty of financial districts such as The Docklands, La Defence or the entire city of Frankfurt with quite tall buildings. Both these skyscrapers and those in North America are far smaller than the latest Asians one. Speaking for Europe the reason for this lies partially in the cultural perception where a skyscraper of Burj Khalifa proportions would be unfitting and economic circumstances where there is no need nor money for such projects.

As a last point I'd like to highlight that though the US was the major builder of skyscrapers in the late 19th and most of the 20th century most of the important innovations that enabled the building of such skyscrapers came from western Europe. A German invented the first reliable elevator and an Englishman made the building with a steel frame viable. If I recall correctly the software to make reliable pressure calculations was made by an Indian who later emigrated.

My sources for this are the lectures of my teacher who gave a course on this subject.