The prime minister's chief adviser Dominic Cummings has said he does not regret driving 260 miles from London during the coronavirus lockdown.
He revealed he had not told Boris Johnson when he decided to take his family to County Durham after his wife developed Covid-19 symptoms.
Mr Cummings said he believed he had acted "reasonably" and within the law.
Mr Johnson said he understood "the confusion, anger and pain" felt and people "needed to hear" from his aide.
He added that Mr Cummings had acted "reasonably" and with "integrity and care for others", but Labour and the Liberal Democrats accused both men of double standards.
Meanwhile, the government said the number of deaths among people who have tested positive for coronavirus, in all settings, had risen by 121 to 36,914.
Mr Cummings has faced several days of attacks in the media, with many people, including some Conservative MPs, calling for him to go.
Speaking in a press conference in the Downing Street rose garden - requested by the prime minister - he said he wanted to "clear up the confusions and misunderstandings".
He added that, despite days of criticism in the press, he had not considered resigning, saying: "I don't regret what I did."
During his statement, Mr Cummings revealed he had:
- travelled from work in Westminster to his home on 27 March after his wife - the journalist Mary Wakefield - fell ill, then returned to work hours later
- driven his wife and son from London to County Durham that evening
- stayed at a cottage on his parents' farm
- developed Covid symptoms on 28 March
- driven on 3 April to pick up his wife and son from hospital, where his son had stayed overnight after he "threw up and had a bad fever"
- driven around 30 minutes from his family farm to the town of Barnard Castle on Easter Sunday (12 April) - 15 days after he had displayed symptoms - in an effort to test his eyesight and readiness to drive back to London
- taken his family back to London on 13 April
Many people, including some Conservative MPs, have called for Mr Cummings to be sacked for making his car journey just four days after the lockdown started, while Labour said he had "clearly broken the rules".
But Mr Cummings told reporters: "I don't think I am so different and that there is one rule for me and one rule for other people."
He added: "I don't regret what I did."
When he found out his wife was ill on 27 March and after "briefly telling some officials in Number 10 what happened", Mr Cummings said he "immediately left the building, ran to my car and drove home".
After a couple of hours she "felt a bit better", he said, and "there were many critical things at work, and she urged me to return in the afternoon and I did".
Mr Cummings said he realised the family would have been left without childcare in London if, like his wife, he had fallen ill, so they decided to drive to County Durham that evening.
BBC Newsnight's policy editor Lewis Goodall tweeted that the "crux of the issue" is whether he "abused" the guidelines by doing so.
Crux of issue hasn’t been answered— Lewis Goodall (@lewis_goodall) May 25, 2020
If a couple of the Cummings enormous resources and social capital couldn’t follow the guidelines in the way everyone understood them without travelling 260 miles, were the guidelines absued in first place? And why could everyone else do it?
Dominic Cummings' rose garden confessional was a bold move designed to take the drama out of a crisis.
But giving detailed answers to why he at the very least broke the spirit of the lockdown rules does not answer the fundamental question now - is his continued presence in Downing Street more of a hindrance than a help to Boris Johnson?
Tempers may have cooled slightly on the Conservative backbenches, but there are still calls for him to go, both private and public.
And some senior Conservative MPs are still aghast at how much political capital the prime minister has burned through to keep Mr Cummings at his side. Opposition leaders still intend to push for his departure.
The man respected by Mr Johnson for judging the public mood has made himself famous for falling foul of that opinion.
His explanations may ease for some of the anger. But in Westminster and beyond, it will not disappear overnight.
And when the prime minister is interrogated by senior MPs on Wednesday his decisions over Dominic Cummings will surely be on the list.
He said his sister and his nieces, who live on his parents' land, had offered to look after his four-year-old son if necessary.
Mr Cummings himself became ill the day after arriving in County Durham, with symptoms including a headache and fever.
He said he had isolated in a cottage around 50 metres from his parents' home but did not have any contact with the couple, in their 70s, other than shouted conversations.
He added that, while he had not informed the prime minister - who himself caught coronavirus - before he drove north: "I did actually speak to him later but neither of us can remember what was said because we were both in pretty bad shape."
Mr Cummings also said his son had suffered a "bad fever" on 2 April. He had gone to hospital by ambulance but had not tested positive for coronavirus, and Mr Cummings had picked him up by car after an overnight stay because there were "no taxis".
Asked about his trip to the tourist hotspot of Barnard Castle on 12 April, which included walking "10 to 15 metres from the car to the riverbank", he replied: "I wasn't sightseeing."
He said: "My wife was very worried, particularly as my eyesight seemed to have been affected by [Covid-19]. She did not want to risk a nearly 300-mile drive with our child, given how ill I had been.
"We agreed that we should go for a short drive to see if I could drive safely."
However, John Apter, chairman of the Police Federation of England and Wales, tweeted that anyone concerned about their vision should not drive in order to "test" their ability to do so.
"It's not a wise move," he wrote.
Mr Cummings insisted he had not stopped during the journey from London to Durham but may have pulled in on the return to London to get petrol.
He had had to stop so his son could go to the toilet in a woods by the side of the road, he added.
And he believed he had kept to government guidelines, which tell people who develop Covid symptoms to stay in their homes, because they allowed for some leeway in "extreme" circumstances.
He said he was not surprised that many were angry about his actions but it had been "a complicated, tricky situation".
Commenting on Mr Cummings' appearance before the media, Mr Johnson said: "To me, he came across as somebody who cared very much about his family and who was doing the best for his family."
But Labour deputy leader Angela Rayner said the hour-long Downing Street press conference had been "painful to watch".
"He clearly broke the rules," she said. "The prime minister has failed to act in the national interest. He should have never allowed this situation with a member of his staff."
A spokesman for the Labour Party added those hoping for an apology had "got none".
Acting Liberal Democrat leader Sir Ed Davey urged Mr Johnson to sack Mr Cummings, adding: "His refusal to have the decency to apologise is an insult to us all. It reveals the worst of his elitist arrogance."
SNP Westminster leader Ian Blackford MP echoed that sentiment, saying Mr Johnson had "no option" but to sack Mr Cummings, and his failure to do "is a gross failure of leadership".