Funding rates are low in many (all?) fields of science in the US, but I agree with @arcanus that some data showing US researchers moving to Europe is warranted.
In my field (astronomy), jobs in the US are tough to come by, but jobs in Europe are generally even more difficult to get. There seem to be fewer jobs in Europe and those jobs are often reserved for people who are already "in the system" and worked at the hiring institution. So many of those jobs to go people who've put in their dues by hanging around, rather than hiring the best people.
Europe does have large European Research Council (ERC) grants which are typically a few million Euros per award. If you win one of those, you're good to go and can fund a sizable research group (at least in astronomy) and can probably use that as leverage to get a faculty job. But those grants are tough to come by.
I can't speak for other fields, but the vast majority of people I've met in astronomy that are moving to Europe are doing so because they're from Europe and want to head home. But the availability of research/faculty jobs there is not good, compared to the US.