A six-mile (10km) exclusion zone has been paced around the farm in the village of Nafferton, within which the movement of all poultry is banned.
The flu strain has been identified as the H5 virus but not the H5N1 strain which can be deadly to humans.
Environment Secretary Liz Truss told the Commons the risk to public health was very low.
Defra's Animal and Plant Health Agency said culling of the livestock was expected to last all day.
Chief Veterinary Officer Nigel Gibbens said the farm had good bio-security in place so the risk of spread was "probably quite low".
The European Commission said the outbreak was likely to be linked to migratory birds - possibly swans - heading south for winter, and to cases in the Netherlands at the weekend and Germany earlier in the month.
The transport of poultry and eggs has been banned throughout the Netherlands after an outbreak of the H5N8 bird flu strain was confirmed at a chicken farm in the central province of Utrecht.
The commission said the "information available indicates that the H5 virus in the UK is probably identical to the H5N8... virus found in the Netherlands and in Germany".
Officials say the H5N8 strain is very dangerous for bird life and could potentially spread to humans, although people can only be infected through very close contact with affected birds.
Prof John Oxford, emeritus professor of virology at the University of London, said the risk to humans was not a major concern.
He said: "There've been outbreaks of this infection in Korea and south-east Asia over the last year. There's been an outbreak in Germany, now an outbreak in Holland, now an outbreak in England.
"In no case, in none of those countries, involving many hundreds of thousands of birds, has there been any serious human infection so I think we can quite safely say why should England be any different, and be a little relaxed about it."
Farmer John Wright, who has more than 1,000 turkeys 20 miles from the affected farm, said he was hopeful the outbreak could be contained.
He told the BBC: "I'm just crossing my fingers and hoping it's just an isolated outbreak as others have been in the past, and that the government will keep on top of the job and do a good job as they have done in the past.
"Of course we're worried; we always are when somebody mentions bird flu, it puts a bit of a fear through you, but I'm sure that it'll be contained and everything will blow over. I'm just hoping anyway."
A private vet reported a possible case of bird flu at the Nafferton farm on Friday morning and the disease was confirmed the following day.
On Monday, specialists dressed in blue protective overalls and face masks could be seen on the farm.
Parish council chairman Gary Lavis said concern was raised about a week ago when egg production began dropping and the number of birds dying increased.
In her statement to MPs, Ms Truss said "immediate and robust action" was being taken to control the outbreak and prevent any potential spread of infection.
"Importantly, the chief medical officer and Public Health England have confirmed the risk to public health is very low," she said.
"The Food Standards Agency have said it does not pose a risk for food safety for UK consumers. The chicken and turkey people eat continues to be safe."
The case is the first in the UK since 2008 when chickens on a farm in Banbury, Oxfordshire, tested positive for bird flu.
A cull of 6,000 ducks is due to start following a confirmed case of bird flu at a breeding farm in East Yorkshire.