The US has actually been testing more people per capita than South Korea (or, indeed, most countries) for a while now - if I remember rightly, for about a month in terms of testing rate and a week or so in terms of total tests done. South Korea is running at about 6,000 tests per day right now with a peak capacity of about 20,000 last used a month or two ago. If scaled up to the US population that would correspond to about 40,000-125,000 tests per day. The US was stuck around 130,000 tests per day for a while but has been at around 200,000 per day for the last few days.
The US media has just carefully worked to convince people of the opposite without technically saying as much, probably for stupid partisan political reasons. South Korea's positive test rate is low because they have few detected cases, and it's not actually clear why - social distancing definitely played a major part, since they had to impose stricter social distancing rules to actually get the number of confirmed cases to drop to the point they are now, but they've dropped those now and even outright lockdowns haven't had the same level of effect elsewhere.
That's become true recently (we're now neck and neck with South Korea), as Korea's case rate has plummeted; South Korea has essentially weathered this storm, or at least the first wave of it. But it's nowhere close to true compared to Italy; our per-capita test rate is half that of Italy's. And Italy's test positive rate is something like half that of ours; the point 'jcranmer makes --- that you can work out whether we're undertesting from our test positive rate --- stands.
The US media has just carefully worked to convince people of the opposite without technically saying as much, probably for stupid partisan political reasons. South Korea's positive test rate is low because they have few detected cases, and it's not actually clear why - social distancing definitely played a major part, since they had to impose stricter social distancing rules to actually get the number of confirmed cases to drop to the point they are now, but they've dropped those now and even outright lockdowns haven't had the same level of effect elsewhere.