Probably depends on where you go.
I don't know about "the US", but as a "European" I thought serving sizes were comparable to what I get in restaurants at home. Drinks were an exception, since basically all restaurants had unlimited soda for next to nothing. This was actually great, since I was riding a motorbike in the desert in July.
For reference, I live in France and visited LA and random towns in the western states.
I'm a big fan of European serving sizes compared to U.S. for food – but when it comes to beverages, particularly water, I can't believe how much they charge you for how little they give. I understand everything comes in bottles with VAT but even asking for tap water I found they'd only bring a very small glass.
I'll be visiting France soon, so will be able to compare on that front. But I think it is an understatement to say that things are universally smaller.
And on the drinks, even places in Japan that had free refills still gave, at largest, an 8oz cup. Usually, I think they were even smaller. Even getting popcorn at Universal, the bags were large, but nothing compared to what I'd expect over here.
Some of this, I'm sure, is having gotten used to ordering the larges. For a time, it was not unheard of to get a 32oz soda at any given convenience store. May still be normal? I don't know.
(And, of course, this isn't getting in to the sizes of vehicles.)
It absolutely varies a lot within Europe too, but my feeling at least is that the difference between European and US portion sizes gets bigger as you move towards low-end places. High-end restaurants are pretty similar in portion sizes almost everywhere I've been, presumably because they're not competing on portion sizes, while lower-end places are much more susceptible to local expectations of what is good value.