The department said the turkeys had tested positive for the H5 strain of the disease. It was not yet known whether it was the deadly H5NI strain, which has killed dozens of people around the world.
The Department for Environment, Food and Rural Affairs said a 3 kilometer (2 mile) protection zone had been set up around a farm in Diss, about 160 kilometers (100 miles) northeast of London, and all 5,000 turkeys, geese and ducks on the premises would be slaughtered.
In February, an outbreak of H5NI bird flu on a poultry farm in the same part of England led to the slaughter of almost 160,000 turkeys.
Britain's first case of H5NI flu was in a swan in Scotland in 2006.
Bird flu has killed or prompted the culling of millions of birds worldwide since late 2003, when it first began ravaging Asian poultry stocks. It has killed at least 205 people worldwide, but remains hard for humans to catch. Experts fear it could mutate into a form that spreads easily among people, potentially sparking a global pandemic. So far, most human cases have been traced to contact with infected birds.