The Health Minister Vaughan Gething AM has told ITV Cymru Wales he is "deeply sorry" after two healths boards in a week postponed planned operations at a combined seven hospitals.
Betsi Cadwaladr have postponed some "routine operations" after Hywel Dda made the decision on Monday 6.
Planned operations at Hywel Dda have been cancelled for a third day in a row.
Hospitals affected include Glan Clwyd, Wrexham Maelor, Ysbyty Gwynedd, Bronglais, Glangwili, Prince Philip and Withybush.
The Health Minister said "I'm deeply sorry that anyone has had planned care that has not taken place when it was due to because that is a level of anxiety for most people".
But Mr Gething AM said the decision to postpone planned operations had been done in order to "prioritise our sickest patients first".
He would not be drawn on when planned operations would return to normal.
Challenged over annual winter pressures being cited for health board crises, Mr Gething said the health service in Wales had learnt lessons from previous years.
Betsi Cadwaladr have issued a statement saying some operations have been postponed "as part of a planned approach to focus our resources on caring for more patients who need to be admitted for emergency assessment and treatment."
The news comes after Welsh A&Es and the ambulance service had their worst ever performance. Latest figures show targets for emergency care were missed across the board in November 2019. Almost 6,000 patients spent 12 hours or more in an emergency care facility - the highest number on record.
On Monday, the First Minister of Wales insisted the plans for hospitals and health boards to cope with pressures over the winter period were "holding up."
A spokesperson for Betsi Cadwaladr said routine operations at Glan Clwyd Hospital were cancelled on Monday and Tuesday but "will be rescheduled as soon as possible."
Hywel Dda have also confirmed all planned inpatient operations have been cancelled on Wednesday "in the interest of patient safety". They have also apologised and said that emergency surgery, day case procedures and outpatient appointments at the four hospitals affected are continuing as normal.
Patients affected have been contacted directly.
In light of the cancellations, Shadow Health Minister Angela Burns AM said she is "appalled" at what she calls a "flippant" reaction from the First Minister.
She said Mark Drakeford AM dismissed the cancellations by saying "only a few" people were affected.