Tests on Muhammed Ozcan, whose sister Fatma died on Sunday at the Van University Hospital, came back positive, the ministry said in a statement.
"The preliminary tests on Fatma are negative, however," said the document, which added that other tests would be carried out.
Fatma's case recalled that of another child, 11-year-old Hulya Kocyigit, one of three siblings from the eastern town of Dogubeyazit to have died from avian influenza.
Experts initially said Hulya's brother and sister, aged 14 and 15, had tested positive for H5N1, but she had not.
But further tests revealed bird flu as the cause of her death several days later.
Muhammed became the 19th person, including the three who have already died, to be infected by the disease in Turkey and outside eastern Asia, where nearly 80 people have died since 2003.
Serious but stable
Muhammed is in "serious but stable condition" and, unlike his sister before she died, is not on an artificial respirator, the hospital's chief physician Huseyin Avni Sahin said.
Dr Sahin described Muhammed as suffering from complications and in a condition "worse than that of others being treated for the same symptoms."
Dr Sahin said the children had been in contact with infected chickens and it was already "late" when they were brought to the hospital on Wednesday, six days after they began showing the first symptoms.
"Fatma was already in bad shape when she got here," Dr Sahin said.
He told reporters that Fatma had butchered infected chickens herself, cooked them and served them to her family.
Eating contaminated chicken does not transmit the disease, but handling them does, experts said.
Experts say it is vital that suspected bird flu sufferers be rushed to hospital as soon as the symptoms are seen.
At least three children, two in Van and one in the northern city of Samsun, who tested positive for the virus but were rapidly hospitalised were successfully treated and sent home last week.
In Istanbul, doctors at a hospital in Goztepe, on the Asian side of the city, said they had discharged a five-year-old boy from Gebze, 100 kilometres to the east, who was suffering from high fever and had been in contact with chickens kept in the yard of his house.
The boy tested negative for bird flu and was sent home, they said.
Non-human cases of bird flu have already been reported in the Istanbul area, but all tests conducted on people there so far have proven negative.
Dr Sahin said the Ozcan siblings, like almost all patients who contract the disease, were being treated with Tamiflu, an antiviral drug considered to be the most effective against avian influenza.
Strikes children
The disease has so far struck at children rather than adults: of the 19 cases registered in Turkey so far, all but two have been children or teenagers, according to health ministry figures.
The current outbreak emerged in Dogubeyazit in late December and spread across the country like wildfire.
It has now been reported in nearly a third of Turkey's 81 provinces where teams of veterinary experts have culled some 600,000 birds so far, according to the agriculture ministry.
Experts from the World Health Organisation said last week that one of the cases in Turkey showed that H5N1 had mutated into a form that transmitted more easily from birds to humans than from birds to birds.