People seeking drive-up coronavirus tests in Phoenix faced a three-mile-long car line last weekend. On Friday, Arizona’s largest laboratory received twice as many samples as it could process. The phone line for testing appointments at a large site on the state fairgrounds now opens at 7 a.m. with 800 callers already in the queue.

By 7:07 a.m., all 1,000 appointments for the day are typically taken.

“We are literally overwhelmed with the numbers requested,” said Dr. Marjorie Bessel, chief clinical officer at Banner Health, the hospital system that runs the fairground site. “The testing is very popular, and very needed, but we don’t have enough of it.”

The United States’ coronavirus testing capacity has begun to strain as the pandemic continues to spread, with over 35,000 cases recorded Tuesday. Across the country, more than a dozen public laboratories say they are now “challenged” to meet the demand.

The problem has become especially acute in Arizona, where rapid spread of the virus has left health care providers and medical labs no longer able to meet testing demands.

Inadequate testing capacity has hampered the American coronavirus response since the start of the pandemic. When the federal government distributed faulty test kits in February, states were unable to monitor the disease’s early spread.

Since then, no national testing strategy has emerged. Local governments and health providers largely decide where to offer testing. And the bottlenecks today are strikingly similar to those in the pandemic’s early weeks: laboratories unable to obtain the machines they need to run more tests, scrambling to hire enough workers to staff them, and a fragmented laboratory system that makes it hard for hospitals and doctor’s offices to coordinate with facilities that could handle excess volume.

American labs continue to compete with one another as well as those abroad for testing supplies like swabs used to collect samples and the machines that process the material.

“The global supply for high-volume instruments has not been able to keep up with global demand, and that is a critical factor to increase testing capacity,” said Julie Khani, president of the American Clinical Laboratory Association. “I don’t think there is any laboratory that has an abundance, or stockpile, of any type of supplies right now.”

The surges in cases happen so quickly — some labs have seen their demand double or triple in a matter of days — that health providers have little time to broker relationships with new testing partners.

“I’m continuously frustrated that this is still a problem,” said Kelly Wroblewski, director of infectious diseases at the Association of Public Health Laboratories. “We should be at a place where getting a test isn’t as challenging as it used to be.”

National testing capacity has expanded significantly since the start of the pandemic, recently reaching half a million daily tests. Federal health officials testified at a congressional hearing this week that the country has the capacity to do 15 million coronavirus tests per month; they expect that number to reach 40 million to 50 million by the fall. Experts have estimated that at least 500,000 coronavirus tests daily are what the country needs to safely reopen.

But that target is a nationwide figure, and it does not account for extra testing that states and cities need to manage large outbreaks. No coordinating entity exists to help overwhelmed labs find extra capacity elsewhere.

Testing demands have grown with each day as states reopen, with employers looking to check workers who are back on the job. Some states now require certain health facilities, such as nursing homes, to regularly test their employees. Local governments are setting up new contact-tracing units that will also require ready access to tests.

Federal officials have offered mixed messages about what testing capacity is needed. The Trump administration recently phased out support for some federally funded testing sites, transferring control to the states. At a rally last week in Tulsa, Okla., President Trump said he had asked to “slow the testing down” because it was increasing the number of confirmed cases. Anthony Fauci, director of the National Institute of Allergy and Infectious Diseases, testified before Congress on Tuesday that he had not received any instructions to reduce testing. Instead, he told legislators that “we need to do much, much more surveillance testing.”

Ms. Wroblewski’s organization has run regular surveys of its members to measure their testing capacity. Of the 88 laboratories that responded last week, 13 said they faced challenges meeting demand.

Those struggling to keep up, Ms. Wroblewski said, tended to be labs serving areas facing a new and large outbreak. “The nature of public lab testing is that they tend to look into outbreaks,” she said. “There are cases of increased demand, and it can take a week to adjust, to pull the staff and change the work flow.”

Testing has also become more challenging as temperatures rise. Many health providers run outdoor drive-up sites to lower the risk of disease transmission. That becomes less feasible in the heat, and some testing sites have had to close.

All of these challenges have become acute in Arizona, which has gone from reporting several hundred daily cases last month to 3,000-plus some days this week. The state recorded its highest number of coronavirus hospitalizations on Monday.

Sonora Quest, the state’s largest medical laboratory, received more than 12,000 coronavirus samples last Friday — twice as many as it can process in a day. “This is not a position we want to be in,” said Sonya Engle, the laboratory’s chief operating officer.

For the past two months, Sonora Quest operated with ample capacity: It could run about 5,000 tests daily, but typically the demand was for only 2,400. Testing demand grew quickly this month as the disease spread.

The lab increased capacity to 6,000 daily tests by running machines all day, every day. It would like to go even higher, but the new testing machine it ordered in May won’t arrive until July at the earliest.

“The suppliers are doing as much as they can,” Ms. Engle said. “But demand is exceeding their ability to deliver at this time.”

Many of Sonora Quest’s tests come from large drive-through sites that have not met patient demand. Equality Health, a network of medical clinics in Phoenix, had overwhelming demand at its drive-through testing event last Saturday. The clinic planned to test about 500 people but 1,000 showed up. Some had to be turned away when the site ran out of test kits.

“This is something that is usually done by health departments, not small health clinics,” said Dr. Edmond Baker, Equality Health’s medical director.

Equality Health will host another drive-up testing event this weekend. It closed registration Monday evening — five days before the event — after receiving 1,142 sign-ups.

Banner Health, the state’s largest hospital system, used to run five drive-through testing sites. In the last few weeks, as temperatures surpassed 100 degrees, it has closed all but one; it became untenable to have workers collect samples all day outdoors. Banner Health has closed four sites and moved all testing operations to the state fairgrounds, where workers in fan-cooled tents see 1,000 people a day.

The hospital system has space on the fairgrounds to see more patients but is limited by the availability of test kits. The Banner Health testing site takes about a week to return results. Dr. Bessel, the chief clinical officer, worries that those lags contribute to further spread of the disease.

“One of the downsides of long turnaround times is that some patients may start feeling better, and assume they’re negative,” she said. “They might make the run to the grocery store or get a coffee with a friend, when they really should not be doing that.”

Results for hospitalized patients, which are prioritized, have also slowed and can take more than a day to come back. That interferes with treatment and can cause health workers to use more of their scarce protective equipment.

“We can’t start convalescent plasma or remdesivir if your test is still pending,” Dr. Bessel said. “I don’t want to paint a picture of nothing being done, but having a test result helps move care along.”