Some 2,257 people were reported in hospital with Covid on Wednesday, up 36 from the day before.
Some 12,421 new cases were reported on Wednesday, an increase on the previous day, while 39 deaths were reported within 28 days of a positive Covid test.
But the number of patients in intensive care has dropped, to 25 from 29 on Tuesday.
NHS Greater Glasgow and Clyde said the situation is “as serious as it gets”, and urged people to stay away from from accident and emergency unless their condition is “very urgent or life threatening”.
Medical director, Dr Scott Davidson, said: “It’s two years since the first national lockdown and we’re still very much in the grip of Covid.
“Our hospitals are almost at capacity and our A&Es are extremely busy.
“This is as serious as it gets. Our teams are under significant pressure and we need the public to show support by only attending A&E if your condition is very urgent or life threatening.”
Anyone who attends at hospital should wear a face covering, he added, as “Covid is rife within our communities and we need to protect patients and staff as much as possible.”
Health secretary Humza Yousaf acknowledged Scots were “suffering” as a result of the current pressure on hospitals.
Speaking on BBC Radio Scotland’s Good Morning Scotland programme on Wednesday, he said: “Every single health board including Greater Glasgow and Clyde are having to deprioritise other treatments, and at times quite urgent treatment, and people are having to suffer as a result.”
However he insisted the country is in a far better place than it was two years ago when the first national lockdown was imposed, citing the impact that vaccination and the development of anti-viral treatments have had in the fight against the virus.
“We are in a much better position when dealing with high case numbers than we were two years ago,” he said.
He added: “I think we need probably a few more days of data to tell us exactly where we are, but if case numbers remain where they were yesterday for example, [if] that pattern remains over the course of the week, I think we could say we are seeing at least a stabilising picture.”
On the two-year anniversary of the first Covid lockdown, NHS Tayside thanked staff for their “amazing achievements”, and recognised “how difficult and challenging it has been at times, both professionally and personally”.
NHS Tayside chair Lorna Birse said: “The commitment and dedication shown by teams across hospitals, community care, GPs and volunteering has been extraordinary.
“The Team Tayside ethos of working together to find new ways of providing care, delivering solutions and innovating, be it through new services or equipment or treatment, has shone through.”
And she added: “It is that determination to adapt and succeed that will see us through as we move towards the next phase of recovery.”