All 10,500 birds at the farm in Northallerton will be humanely culled to limit the spread of the disease.
A temporary control zone has been put in place around the infected site.
Public Health England (PHE) said the risk to public health from the virus was “very low” and the Food Standards Agency said the disease posed a “very low food safety risk”.
The government said there was not anticipated to be any impact on the supplies of turkeys or other birds over Christmas.
Christine Middlemiss, the government’s chief veterinary officer, said “immediate steps” were taken to limit the risk of the disease spreading.
“Public Health England has confirmed that the risk to public health is very low and the Food Standards Agency advises that bird flu poses a very low food safety risk for UK consumers,” she said.
“Bird keepers should remain alert for any signs of disease, report suspected disease immediately and ensure they are maintaining good biosecurity on their premises.
“We are urgently looking for any evidence of disease spread associated with this farm to control and eliminate it.”
Dr Gavin Dabrera, consultant in acute respiratory infections at PHE, said: “To date the World Health Organisation has never confirmed any cases of H5N8 in humans and the risk to the public is considered very low.
“As a precaution the local Health Protection Team will offer routine health advice to those working on the farm. We will work with Defra [the Department for Environment, Food and Rural Affairs] to monitor the situation closely.”
A Food Standards Agency spokesperson said: “We advise that, on the basis of the current scientific evidence, avian influenzas pose a very low food safety risk for UK consumers.
“Properly cooked poultry and poultry products, including eggs, remain safe to eat.”
A detailed investigation is in progress to determine the most likely source of this outbreak.
Wild birds migrating from mainland Europe during the winter period can spread the disease to poultry and other captive birds.