So. Korea tested early. The US is just starting large scale testing.
So I think that gives the false impression that the number of new US cases is quickly increasing. Meanwhile in So. Korea, where the testing has been ongoing, (I think) they are on a flat or downward trend of new cases.
If you test more you find more cases. That increases the count.
But, as you point out, testing early as in South Korea has helped slow the spread of the virus. But that is because they managed to find new cases early and then isolate these people. They immediately acted to reduce the number of people capable of infecting others.
And that doesnt reduce the count but it can radically slow the growth rate of new cases. Slow that growth enough and the threat will rapidly fade away.
So I think that gives the false impression that the number of new US cases is quickly increasing.
Institution of widespread testing will indeed pick up pre-existing cases that were just floating out there, but that effect will stop after an initial surge. By the end of the week, that surge - to the extent it exists more than trivially - from this cause will largely be over.
Most of the cases coming forwards are still not community-spread. They may have come from someone who is not linked to a known previous case, but most of them can be traced back.