"bad headache" is correct and usable in written English.
You could use it in a sentence such as: "I had a bad headache yesterday so I stayed in bed all day."
I had a very, very bad headache.
I'm the sort of person for whom a bad headache can never just be a bad headache; it must be the onset of a brain tumor.
Everyone who got into the office today had bags under their eyes and a bad headache.
"They told me I had a really bad headache," he said.
Death, a stroke or, at minimum, a very bad headache would have ensued.
He had such a bad headache that he sent the drummers home once and for all.
A heavily pregnant woman walks into the surgery with a bad headache.
Posada, who told a team spokesman that he had a bad headache, declined to speak to reporters.
Vonn said she went to bed Tuesday with a bad headache but woke the next day feeling better.
With a bad headache at work in the hospital, I went out at lunch to buy painkillers – forgetting it would take me over my daily budget.
With the encephalitis I woke up one day with a bad headache, feeling terrible and I gradually lost my motor skills until I ended up in intensive care.
