'do you often suffer from nosebleeds?' is a very natural translation for this, I think.
What does blør mean on its own (apart from "have")?
I just want to point out that å blø does not mean to have. This is an expression: blø neseblod – have a nosebleed. You may also have noticed other pain-related expressions like ha vondt i hodet – have a headache.
To respond to your question (someone unfortunately deleted their answer): å blø means to bleed (lose blod).
So this is then a cognate object construction, like we have it in English as well (e.g. to die a painful death)? Thanks for the explanations!
Yes. I'd love to be able to list others in Norwegian, but I can't think of any at the moment.