Hospitals in West Yorkshire have warned patients only to attend accident and emergency in a "genuine life or limb-threatening" situation.
Bradford Teaching Hospitals NHS Trust and Airedale Hospital both shared a statement on social media saying healthcare services were under "extreme pressure".
The statement said: "Only come to A&E if it is a genuine life/limb-threatening emergency so we can care for our sickest patients."
It comes after warnings that the situation in the NHS has become "intolerable", with patients facing long waits for treatment, ambulances delayed, and thousands of beds taken up with medically-fit people who should not be there.
The Royal College of Emergency Medicine has repeated a claim that between 300 and 500 people are dying each week as a result of delays and problems with urgent and emergency care.
More than a dozen NHS trusts and ambulance services across the UK declared critical incidents over the festive period, with officials saying rising flu cases and the impact of the COVID-19 pandemic have hit the health service.
Last week GPs in West Yorkshire were told to "step down" routine activity to prioritise urgent cases because of "extreme" pressure on hospitals.
In a letter to GPs the NHS West Yorkshire Integrated Care Board said: "The whole health and care system is currently experiencing extremely high levels of demand, increase respiratory illness and increased staff sickness."
The board warned of widespread "overcrowding, long waits to be seen and delays for people requiring admission".
Health bosses in Leeds have issued an appeal to people with flu-like symptoms to stay away from A and E.
The Leeds Teaching Hospitals NHS Trust said: "Our hospitals continue to experience significant pressures and we have lots of people attending our A&Es with flu-like symptoms."
At hospitals run by Mid Yorkshire NHS Trust, which includes Pinderfields and Dewsbury Hospital, some outpatient appointments have been postponed "due to the unprecedented demand on services".
In a statement, a spokesperson said: "We have taken the difficult decision to postpone those outpatient appointments and elective procedures next week which can safely be delayed. This is so our staff can focus their care on urgent and critical patients."
Meanwhile, Doncaster and Bassetlaw NHS Trust said the local NHS was "under the most strain we have ever seen" and Sheffield Teaching Hospitals warned of "much longer waits than normal for care" in emergency departments.
The Prime Minister's official spokesman has acknowledged that the current pressure facing the health service was an "unprecedented challenge".
They said: "I think we are confident we are providing the NHS with the funding it needs - and as we did throughout the pandemic - to deal with these issues.
"I think we have been up front with the public long in advance of this winter that because of the pandemic and the pressure it's placed in the backlog of cases that this would be an extremely challenging winter, and that is what we are seeing."
He pointed to delayed discharge as a concern but said the Government is doing "everything possible" to increase the number of beds available.
A spokesperson for the Department of Health and Social Care said the NHS and social care have been backed up with £14.1 billion additional funding over the next two years plus an extra £500 million to speed up hospital discharges this winter.
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