Some institutions definitely have this problem much worse than others in the US, and in certain subjects; while my CS courses were generally taught by professors, the CS labs were definitely run by grad students, grad students did all the grading, etc. For niche elective courses that I took to complete my humanities requirements, those were definitely more likely to be taught by a grad student. But YMMV; talking to peers from secondary school, it seemed like certain institutions (Harvard, Berkeley) were more likely to have this problem. Can't really verify that hearsay though.
On the topic of large class sizes, there is definitely room for improvement in the US system; being at a poorly-funded mid-tier public university, we had several courses that were so large that there were overflow rooms with a camera pointing to the main lecturer in the main room, and these were lecture halls that could easily hold 400.