Theresa May has told the public she is "on their side", laying the blame for the delay to Brexit squarely with MPs.
Speaking from Downing Street, the prime minister said people were "tired of infighting and political games" and it was "high time" politicians made a decision on the next steps.
Earlier, Mrs May wrote to EU Council President Donald Tusk requesting to delay Brexit until 30 June.
Jeremy Corbyn said she was "in complete denial about the scale of the crisis".
Mrs May was forced to ask for a postponement after MPs twice rejected the withdrawal deal she has negotiated and also voted to reject a no-deal departure.
She said the delay was a "matter of great personal regret", but insisted she would not be willing to extend Brexit any further than 30 June – despite appeals from some MPs for a longer extension to give time for a change in direction.
The UK is set to leave the EU next Friday, on 29 March, unless the law is changed.
All other 27 EU members would have to agree any extension beyond that date.
Mr Tusk said he believed the EU would agree to a short extension, but only if Mrs May's deal is signed off by MPs next week at a third time of asking.
In her statement, Mrs May said: "Of this I am absolutely sure. You, the public, have had enough.
"You are tired of the infighting, tired of the political games and the arcane procedural rows, tired of MPs talking about nothing else but Brexit when you have real concerns about our children's schools, our National Health Service, knife crime.
"You want this stage of the Brexit process to be over and done with. I agree. I am on your side."
The PM said it was "now time for MPs to decide" whether they wanted to leave with her deal, no deal or whether they chose not to leave at all – the latter, she warned, could cause "irreparable damage to public trust" in politicians.
"So far Parliament has done everything possible to avoid making a choice," said Mrs May. "All MPs have been willing to say is what they do not want."
She made a final appeal to MPs to back her deal and told the public: "You just want us to get on with it and that is what I am determined to do."
By Laura Kuenssberg, BBC political editor
It's not me – it's them.
Theresa May has pitched herself tonight against Parliament and on the side of the people.
It's true that No 10 believes strongly that swathes of the population have simply had enough of Brexit.
The way it drowns out other public concerns, the way its processes, contradictions and clamour have wrapped their way around the normal workings of Westminster – remote at the best of times and downright bizarre at the worst.
But, when it is MPs the prime minister needs to get on side if she is to have a real chance of finally getting her deal through next week – third time extremely lucky – the choice of message was not without risk.
Mrs May will travel to Brussels for a summit of EU leaders on Thursday, where she is expected to discuss the extension with other member states.
In her letter to Mr Tusk, the prime minister said she had wanted to hold another Commons vote on her withdrawal agreement this week but had been prevented from doing so by Speaker John Bercow.
On Monday, he ruled that bringing it back a third time in its current form would break longstanding conventions designed to prevent MPs from being repeatedly asked the same question.
BBC Europe editor Katya Adler says the mood in Brussels is very sombre as there is a feeling that a no-deal Brexit is now a very real possibility.
A lot is riding on Theresa May's address to the summit on Thursday, our correspondent adds, but the PM's past performances have not gone down well.
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Mr Corbyn criticised Mrs May after the speech, saying she was "unable to offer the leadership the country needs".
The Labour leader added: "To continue to bring back her damaging and twice rejected deal without significant changes, while threatening a no deal outcome ruled out by MPs, is unacceptable and reckless."
A string of other MPs also reacted angrily to Mrs May's comments.
Conservative Remain-backer Dominic Grieve said her "attack on the integrity of MPs is very unfortunate".
Speaking to BBC News, the former Attorney General said he would not be "bullied by anyone in government into supporting something that I think will do our country a great deal of harm".
The Prime Ministers statement was disgraceful. Pitting Parliament against the people in the current environment is dangerous and reckless. Yesterday her government attacked their civil servants. Now shes attacking the MPs whose votes she needs. It will have cost her support
— Lisa Nandy (@lisanandy) March 20, 2019
End of Twitter post by @lisanandy
Liberal Democrat MP and supporter of another referendum, Wera Hobhouse, added: "She is not on my side. We will keep fighting for a Peoples Vote."
No @theresa_may you are to blame for the #BrexitCrisis The impasse in Parliament is all of your own making. You never reached out to the 48% & to other parties to create a compromise. You set down your own red lines from the outset. You have let everyone down.
— Anna Soubry MP (@Anna_Soubry) March 20, 2019
End of Twitter post by @Anna_Soubry
And former Ukip leader Nigel Farage tweeted that the PM's speech was "appalling and pathetic", adding: "The Brexit betrayal is hers."
As usual another statement saying very little. PM says she regrets having to delay. Then don't do it! It is in the Prime Ministers power to ensure we leave on the 29th, regardless of what Parliament suggests. She just needs to get on with it and deliver.
— Andrea Jenkyns MP #StandUp4Brexit (@andreajenkyns) March 20, 2019
End of Twitter post by @andreajenkyns
The prime minister met opposition parties to discuss her proposal for a delay on Wednesday evening ahead of her statement, but sources told the BBC that Mr Corbyn walked out and other leaders remained unimpressed with what they heard.
Having spoken to several of those present at meeting with PM, heres what Ive been told happened;
– PM restated the case for her deal
– Didnt entertain alternatives seriously
– Corbyn didnt want to take part with Independent Group
— Nick Eardley (@nickeardleybbc) March 20, 2019
End of Twitter post by @nickeardleybbc
The Labour leader will also travel to Brussels on Thursday to meet the EU's chief negotiator Michel Barnier. He is expected to hold talks with several EU 27 leaders too.
Earlier on Wednesday, Mrs May met a group of about 20 MPs from her own party who voted against her deal in the first meaningful vote, but backed her in the second.
One of them, Eurosceptic Nigel Evans, said she was told "her neck was on the block".
He said MPs told her if her deal went through, she should not be part of the next phase of the Brexit negotiations, adding: "The buck stops with the prime minister."
Meanwhile, an emergency debate took place in Parliament on Wednesday afternoon, with Labour pressing for further detail about the PM's intentions and demanding that any delay is long enough to allow MPs to "break the impasse and find a way forward".