The notions of a twistor space of a contact manifold and a contact connectionon such a manifold have been introduced by L. Vezzoni as extensions of thecorresponding notions in the case of a symplectic manifold. Given a contactconnection on a contact manifold one can define an almost $CR$-structure on itstwistor space and Vezzoni has found the integrability condition for thisstructure. In the present paper it is observed that the $CR$-structure isinduced by an almost contact metric structure. The main goal of the paper is toobtain necessary and sufficient conditions for normality of this structure interms of the curvature of the given contact connection. Illustrating examplesare discussed at the end of the paper.
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