On Monday, President Trump said that the number of coronavirus tests in the United States now exceeds that in other countries, seeming to suggest that the country is doing a better job of monitoring the spread of the disease than those countries. But his statement, during the daily the White House briefing, overlooked the fact that the United States has a much larger population than countries like South Korea, which has been extensively testing its residents. On a per capita basis, the United States had tested far fewer people than several other countries.

South Korea has tested thousands more people for every million residents compared to the United States, according to its government website as well as the Covid Tracking Project, which tracks testing in the United States. According to the project, 1.9 million people had been tested in the United States as of Monday.

Adm. Brett P. Giroir, the assistant secretary of health who is overseeing the government’s testing response, said quick-response tests are being shipped around the country, and that the government is working closely with hospitals and other laboratories to ensure they have what they need. “I think testing is really in a good position right now,” he said.

Federal inquiries have begun to determine how the nation’s testing capacity turned into such a debacle. The Centers for Disease Control and Prevention had manufacturing errors with the first test it devised for public health labs around the country, and so testing in the states stalled as the virus began to spread in Washington State, New York and California. The Food and Drug Administration, charged with approving the test, was so frustrated that the agency pushed for the C.D.C. to stop making it on site and instead send it to Integrated DNA Technologies, an outside lab.

The F.D.A., for its part, was slow to recognize the danger of the pandemic, and how critical testing by commercial labs and hospitals would be as the virus spread.

In early March, the nation’s two largest commercial labs, Quest Diagnostics and LabCorp, started testing, and they have acknowledged that their labs around the country were overwhelmed. Quest’s backlog is 80,000, according to the company, down from 160,000 on March 25. LabCorp says it has caught up, and now has a turnaround of four to five days from pickup.

Supplies of test swabs have gotten so low that most hospitals test only their most vulnerable patients, typically those being admitted.