Can nose sprays cause bloody nose?

Yes: Yes, especially steroid nasal sprays. They cause thinning of the mucosa and skin so it's easier for the underlying blood vessels to get disturbed and tear.

Yes: Nosebleeds are one of the common potential side effects of all nasal sprays.

Yes: They can irritate the tissue and steroid sprays can thin the nasal lining. Both cases result in increased risk of bloody nose.