In simple infection-control terms, if you compare the US response to China's, the US measures pale in comparison. In China entire cities were locked down hard, with people not being allowed to leave their residences (mostly apartments), with only select few delivering food etc. Transport in and out of such cities was almost entirely shut down. Even asymptomatic people were forced into ad-hoc treatment/containment facilities.
(A bit of an aside, or a "natural experiment", Romania provides an example of what happens when people intially forced into treatment/quarantine are allowed to leave en masse; Romania's Constitutional Court overturned an initial government decision in that regard, likely being the cause of a large increase in cases. Their 2nd/August wave being substantially larger than their first/spring, although other Balkan countries have seen a similar pattern, so causality may not be that certain. Other sources attribute the rise (larger 2nd wave) in the Balkans to the [too] rapid relaxation of containment measures in general.)
Lockdowns do work at reducing infection at least in the short run. A more "apples to apples" comparison being Sweden vs its neighbors. By 23 June, Denmark, Norway or Finland had less than a third of Sweden's number of cases, per capita. (Now, in their defense, Swedish officials claim that in long run the total number of cases will "equal out", an assertion yet to be proven.) Besides such cross-country comparisons, there is a micro-level study from Italy on the effectiveness of lockdowns, using phone location data as a proxy measure for lockdown effectiveness at reducing travel/mobility.
Other countries have developed alternative strategies to lockdown, South Korea is one, focusing on testing and rapid tracing. The US pretty much failed to seriously implement any strategy that would have resulted in a substantial reduction in the spread of the infection, aside perhaps from banking on their pharma industry to rapidly produce a vaccine. (A "check" that has yet to be "cashed".) Also, unlike most European countries, the US also "reopened" while its infection rate had not gone down significantly, although in practice what this meant was that US states that were not hit hard in the initial wave failed to learn from the experience of those that were, reaching and surpassing their infection rates. (And speaking of history repeating itself, South Korea was not exactly immune to this either as a large August church rally was linked to hundreds of new cases, reminiscent of the early days of the pandemic in that country.)
One thing that you might have expected the current US administration to be good at was imposing travel restrictions, which would have bought the US more time. Alas, even in that department, a CDC study says that travel bans from Europe came too late.
Assigning political blame for these facts is beyond the scope of this post.