The UK must go "further, faster" to ramp up its testing capacity for coronavirus, Cabinet Office minister Michael Gove has said.
The government has set a target of carrying out 25,000 tests a day - but that will not be met until the end of April.
Mr Gove said there was a global shortage of the chemicals needed to test patients.
More than 8,000 patients were tested on Monday.
Speaking at the government's daily coronavirus briefing, Mr Gove said: "More NHS staff are returning to the front line, and more testing is taking place to help those self-isolating come back, and to protect those working so hard in our hospitals and in social care.
"But while the rate of testing is increasing, we must go further, faster."
He added that a "critical constraint" on the ability to increase testing capacity was the availability of the chemicals needed to test patients.
"We are working with companies worldwide to ensure that we get the material we need to increase tests of all kinds," he said.
His comments come amid mounting criticism about the inability of some health staff to get tests.
The Royal College of Physicians has said as many as a quarter of doctors are off work because they are having to self-isolate - either because they are showing symptoms or a member of their household is.
A fifth of nurses have been affected, the Royal College of Nursing said, while the British Medical Association said staff began being tested over the weekend, but only in low numbers.
BMA chairman Dr Chaand Nagpaul said: "It's been well over two weeks since the government said it was going to roll out priority testing for healthcare staff.
"But many doctors still have no idea about where or how they can get tested."
Meanwhile, an A&E nurse who did not want to be named told the BBC: "Staff are not being tested, and protective clothing is limited.
"It does feel unnerving. Our team is very strong at work and our matron has said no staff are to go near patients if we don't have the protective clothing.
"The protective clothing comes in daily but we are going through it so fast that it becomes limited by the end of the day."
Later in the briefing, Mr Gove said the UK was working to secure more ventilators, including buying them from EU nations and placing orders at home.
The first "of thousands" of new ventilator devices would roll off the production line this weekend and be delivered to the NHS next week, he said.
It comes as the biggest daily increase in the number of people who have died with coronavirus in the UK - 381 - was reported.
As of 17:00 BST on Monday, the total number of UK deaths linked to the virus was 1,789.
But NHS England medical director Prof Stephen Powis said there were still some "green shoots".
He said there was a "bit of a plateau" in the number of new cases, despite the total rising to more than 25,000 cases overall.
Earlier in the epidemic the number of cases was doubling every three days, but the rate of increase is lower than that now.
These are largely just the cases diagnosed in hospital as people with mild illnesses in the community are not being tested.
But Prof Powis added: "It's really important not to read too much into this.
"It's early days, we're not out of the woods, we're very much in the woods.
"The number of infections is not rising as rapidly as it once was.
"So green shoots, but only green shoots and we must not be complacent and we must not take our foot off the pedal."
He also said there was still "headroom" in the NHS to treat patients despite the rising number of cases.
Some 10,767 people are currently in hospital in England, but Prof Powis said even in London, which has seen the most cases at 3,915, there were still beds free thanks to the increase in intensive care capacity.
The number of beds have been doubled to 1,400 with more due to open later this week when NHS Nightingale - the field hospital at the Excel Centre in east London - starts accepting patients.
Elsewhere in England, the Midlands is also starting to see a rising number of admissions with 1,918 currently in hospital, Prof Powis said.
Mr Gove described the sharp rise in the latest reported UK deaths linked to coronavirus as "deeply shocking" but he could not say exactly when the peak of the epidemic would come.
"There's not a fixed date like Easter when you know that the peak will come, it depends on the actions of all of us," he said.
He added that "now is absolutely not the time for people to imagine there can be any relaxation or slackening" of lockdown measures.
In other developments:
- Anti-malarial drugs are among those being tested in UK clinical trials in a bid to find medicines to help lessen the effects of coronavirus
- A coronavirus app that alerts people if they have recently been in contact with someone testing positive for the virus "could play a critical role" in limiting lockdown
- Pregnant prisoners could be granted temporary release from prison "within days" to protect them and their unborn children from the virus
- The transaction limit for contactless card payments will increase from £30 to £45 per transaction from Wednesday
- British Airways said it would suspend its schedule at London Gatwick
- The UK mortgage market went into lockdown as lenders pulled out of new deals
- The government's cabinet group of senior ministers met entirely through video conference technology for the first time as Prime Minister Boris Johnson continued self-isolation
- Grocery sales for the month of March beat all previous UK records as shoppers stocked up