Theresa May branded the support for Grenfell Tower families in the immediate aftermath of the fire as “not good enough”, with police revealing that 58 people are missing and presumed dead.
The Prime Minister said there had been “huge frustrations” on the ground as people struggled to find information.
She added: “The response of the emergency services, NHS and the community has been heroic.
“But, frankly, the support on the ground for families who needed help or basic information in the initial hours after this appalling disaster was not good enough.”
It came after she met victims of the blaze at Downing Street, amid criticism she had not seen them in the immediate wake of the disaster.
Metropolitan Police Commander Stuart Cundy warned that the death toll could rise further as he formally identified a first victim as 23-year-old Syrian refugee Mohammad Alhajali.
Of the 58, he said 30 were confirmed dead. Sixteen bodies have been taken to the mortuary including one person who died in hospital - the others have been recovered from the building.
Mr Cundy said: “Sadly, at this time there are 58 people who we have been told were in the Grenfell Tower on the night that are missing, and therefore sadly, I have to assume that they are dead.”
“That number 58 may change. I really hope it won’t, but it may increase.
“Our focus has been on those that we know were in Grenfell Tower. However, there may be other people who were in there on the night that others were not aware were there.
“That is also an absolute priority for the investigation - to establish who they may be,” he added.
Mr Cundy also appealed to anyone who may have escaped from the building, but has not yet come forward, to make themselves known.
Of the 58, he said 30 were confirmed dead. 16 bodies have been recovered from the Tower and taken to a mortuary.
Mr Cundy said the police investigation into the blaze would look at the building and its refurbishment in 2016 and vowed to prosecute people “if there is evidence”.
He said: “The investigation is a police investigation. We investigate criminal matters. The investigation will identify any criminal offence that has been committed. It will be wide ranging.
“It will go to establish the answers of what happened in the fire and how it spread, it will look at the building itself, it will look at the refurbishment as well.
“Our criminal investigation will identify any criminal offences that have been committed. Wherever we can, we will bring people to justice if there is evidence. It is completely and wholly inappropriate for me to talk about details of the investigation which may subsequently jeopardise any criminal proceedings.”
Family liaison officers are working with 52 families, and as soon as victims are identified, their loved ones will be told.
Mr Cundy said at the scene: “At this point in time we have 16 people who have been recovered to the mortuary.
“I absolutely understand the frustration of why figures haven’t been released earlier. The reason for that - at one point, in terms of our casualty bureau, there were 400 people who were reported missing from Grenfell Tower.
“Grenfell Tower itself is 120 flats. We have worked tirelessly over the last four days to truly understand those that we know were there on the night.”
He added: “I understand - I really do understand - the frustration of so many about not knowing the scale of the tragedy that is unfolding behind us.
“I have said it before, you have my absolute assurance that as soon as I can possibly tell you something that I know to be accurate, I will tell you.”
Anger flared in the Kensington community over the weekend, as some accused the authorities of withholding information.
Mr Cundy said: “The investigation will be exhaustive. My intention is that it will help provide answers.
“If, as we investigate, we identify issues that are a risk to public safety, we will not be waiting until the end of the investigation before we provide that information to the appropriate authorities.
“If there are any safety issues that we and experts that we will be using identify, we will share that immediately.”
The police are appealing to anyone with pictures or videos of the blaze to hand them in, as they may help establish not only where and how the fire started, but also how it spread.
It came as Prime Minister Theresa May met victims at Downing Street.
The meeting on Saturday comes amid criticism levelled at Mrs May for not meeting those caught up in the fire in the immediate wake of the tragedy.
Mrs May arrived in Downing Street on Saturday where she is chairing the Government task force on the disaster. She will meet victims, volunteers and community leaders afterwards, No 10 said.
A Downing Street spokesman said: “The Prime Minister is this morning chairing a cross-Government meeting to ensure everything possible is being done to support those affected by the Grenfell tragedy.
“Afterwards, she will meet a group of residents, victims, volunteers and community leaders in No 10. The PM has sent her best wishes to HM Queen on the event of her birthday.”
On Saturday, NHS England said 19 people were being treated in hospital, including 10 in critical care.
Search and rescue teams from London Fire Brigade reached the second from top floor inside the Tower on Saturday.
On Friday, grief over the disaster turned into anger as protesters took to the streets to vent over the fire which killed at least 30, with dozens more deaths feared.
Mrs May was greeted with cries of “coward” and “shame on you” as she returned to the site of the devastating fire in west London on Friday.
Later, demonstrators stormed the offices of Kensington and Chelsea Council over its handling of the crisis amid concerns that earlier renovation work was linked to the dramatic spread of the blaze.
Hundreds of protesters also marched on Whitehall, central London, to voice their frustration at the Government’s response to the fire, which ripped through the tower block in north Kensington on Wednesday morning.
Two of the dead have been confirmed as 24-year-old artist Khadija Saye and Syrian refugee Mohammed Alhajali, 23.
Ms Saye was in her flat on the 20th floor when the fire struck, with her mother Mary Mendy, who is thought to be in her 50s.
Tottenham MP David Lammy confirmed the news on Twitter, writing: “May you rest in peace Khadija Saye. God bless your beautiful soul. My heart breaks today. I mourn the tragic loss of a wonderful young woman.”
The Queen and Duke of Cambridge went to temporary relief centres on Friday where they met volunteers and residents who had lost everything.
The council said 110 households had been given temporary accommodation by Friday morning, and added that it was working to find more permanent homes.
But the authority’s latest statement said: “While we will try to do our utmost to ensure those affected remain in or near the borough, given the number of households involved, it is possible the council will have to explore housing options that may become available in other parts of the capital.”
Mrs May’s most senior minister, First Secretary of State Damian Green, defended the way she had handled the tragedy.
Mr Green said suggestions the Prime Minister does not seem to have what it takes to respond to such a disaster were “terribly unfair”.
He told BBC Radio 4’s Today programme: “She’s distraught by what happened as we all are.”
“Absolutely she has the same degree of sympathy and horror at these events that we all have,” he added.
Mr Green said the Government expected to appoint a chairman to lead the public inquiry “in days rather than weeks”.
“We want it to be able to have interim reports as well,” he added. “So this is not going to be one of those exercises of using a public inquiry to delay a response. Actually, we want the response to be as fast as possible.”
Mr Green said the probe will look at whether sprinklers should be retrofitted to tower blocks and the Government will “follow the recommendations of the public inquiry”.