Boosting testing in Wales is "too big" a job for the country's health minister Vaughan Gething to do by himself, the Welsh Conservatives have said.
The Welsh Government has faced criticism for not meeting a target to provide up to 5,000 tests per day by mid-April.
Angela Burns called for one individual to be put in charge of overhauling Wales' coronavirus testing regime.
First Minister Mark Drakeford said Mr Gething is doing "an outstanding job".
But amid complaints that testing in Wales is complex, slow and bureaucratic, Mr Drakeford admitted the system has been too cumbersome and promised to speed it up.
Meanwhile Plaid Cymru leader Adam Price called for the Welsh Government to restart contact tracing as a way to bring Wales out of lockdown.
The Welsh NHS can currently test around 1,300 people per day, but not all that capacity is used. A review has been promised by the end of this week.
Ms Burns, the Welsh Conservative health spokeswoman, said: "I'm not calling for the health minister to resign, but it needs to be one person's job to sort out the bottleneck in testing."
She said: "That person needs to be able to ensure the drive-through test centres are finally up and running, or to see if tests can be completed faster in Manchester or Liverpool rather than driving tests down from North Wales to Cardiff.
"They also need to make sure that the people who can have a test, know it, and know how to access it."
Wales' testing "has been a failure" exacerbated by "inefficiency", Ms Burns claimed.
"I'm sorry to say that this job is just too big for the health minister to do by himself, despite having the resources of the Welsh Government behind him."
Ms Burns told BBC Wales the current testing regime was "far too complicated for everyone to work through, and is bureaucratic and inconsistent", with several different agencies involved including health boards, councils and Public Health Wales.
"You've got patches of excellent practice. You've got patches of diabolical practice," she said, claiming she had heard examples of long delays on getting test results back.
Ms Burns feared Covid-19 would continue to be difficult to manage until vaccines are available and called for "whole community testing".
The Conservative-run UK government is also being scrutinised for its performance on testing - it has aimed to provide 100,000 a day in England by the end of April, but it is not yet anywhere close.
"I'm not responsible for the English NHS," Ms Burns said, but added that if she was an MP "I'd be saying to Matt Hancock very clearly what is going on?"
Responding to Angela Burns' claims, the first minister told BBC Wales Today: "It isn't one person doing the job, this is a cabinet effort here in Wales. We meet every day, we share out the tasks.
"Vaughan Gething I think is doing an outstanding job."
He denied the Welsh Government was failing on testing. Explaining why the target of 5,000 had not yet been met, Mr Drakeford said equipment from overseas had not arrived, and not all regulatory permissions had been granted.
"We will extend the number of tests we have in Wales," he added.
While admitting the testing system had been too cumbersome up to now, the first minister said changes were being made to make it easier for councils to get staff tested.
"It is important to have a system, but we want to slim that system down [and] make it quicker," he said.
'Re-adopt that policy'
On BBC Radio Wales Breakfast with Oliver Hides, Plaid Cymru's Adam Price said he thought testing had "been the major policy failure" for both the Welsh and UK governments.
He said the Welsh Government had identified that and tried to revert to a policy of increasing testing numbers, but criticised ministers for not yet reaching the 5,000 a day target.
"If we are going to come out of the lockdown without seeing a second wave of infection we have to re-adopt that policy that was dropped of testing and contact tracing," he said.
"That is the policy that has been adopted in those countries that have been successful in limiting the levels of mortality."
The chief executive of the Welsh NHS, Andrew Goodall, suggested on Thursday Wales will try to build-up to "community surveillance" for Covid-19 once capacity for testing coronavirus increases.
A Welsh Government spokesman said: "On Tuesday, the health minister commissioned a review of the testing regime to identify where improvements could and should be made.
"The outcome of the review is expected by the end of this week."
He added: "Currently tests take about 24 hours to turnaround, but we plan to speed this up by bringing testing closer to critical workers' homes and opening new testing labs in west and north Wales."