The virulent strain, which has killed scores of people around the world, was discovered Sunday on a farm with about 5,000 turkeys, 1,000 ducks and 500 geese.
As a precaution, officials said they will extend the slaughter to four other farms with a total of 24,000 turkeys.
All five farms are associated with the same company, Gressingham Foods, the Department of Environment, Food and Rural Affairs said, adding that it wants to see if H5N1 has spread among them.
Gas was being used to quickly kill the birds and their carcasses were being taken away in leak-proof trucks.
"We're in the process of trying to find out what the source of this outbreak was," a DEFRA spokesman said on condition of anonymity, in keeping with department regulations.
Bird flu's return to Britain weeks before the Christmas holidays is yet another blow to Britain's farmers, already struggling after livestock herds were hit this year by foot-and-mouth and bluetongue.
The unidentified source of the outbreak was closely related to the strains found in the Czech Republic and Germany earlier this year, Chief Veterinary Officer Fred Landeg said.
If the bird flu spreads, it could devastate the hugely profitable Christmas trade in poultry.
Landeg said Britain had successfully contained an outbreak of H5N1 earlier this year that led to the slaughter of 160,000 turkeys.
Millions of birds worldwide have died or were slaughtered since late 2003, when H5N1 began ravaging Asian poultry stocks. It has killed at least 206 people worldwide since 2003.
Experts believe most victims were probably infected through direct contact with sick birds.
Bird flu is difficult for humans to catch, but experts fear it could mutate into a form that spreads easily among people, potentially sparking a flu pandemic.
In Europe, H5N1 previously was found in France, Switzerland, Germany and elsewhere. Human cases have been recorded in Turkey and Azerbaijan.
The behavior of the Russian inspector satellite, which was launched in the autumn of 2017, puzzles military officials in the United States
When the bill was submitted to Congress on August 2, the reason for imposing the new sanctions on Russia was based on Russia's alleged interference in the US presidential election in 2016, but then something clicked