Hywel Dda director of operations Joe Teape said all hospitals were full following a busy bank holiday weekend.
He said there were a high number of patients who needed to be admitted which put pressure on impatient beds.
Mr Teape thanked staff for "working tirelessly" and asked people to "bear with them at this busy time."
In a statement the health board said additional medical staff were working to ensure "all patients had a senior medical review" both in A&E and on the wards.
Community teams are also at the hospitals to support patients on discharge.
The health board said it had to cancel some "non-urgent planned operations" on Tuesday and will review Wednesday's operations during the day.
"Every winter we deal with a number of seasonal pressures," Mr Teape said.
"To mitigate the demand on services, we have a winter plan to ensure that services continue to run as smoothly as possible while ensuring that patients' needs continue to be met 24 hours a day."
He added that local communities could help by making using of NHS services such as pharmacies rather than visiting A&E departments, where appropriate.
A health board has brought in extra staff and cancelled some operations in an attempt to manage "significant pressures".