Zero-shot multi-speaker text-to-speech (ZSM-TTS) models aim to generate a speech sample with the voice characteristic of an unseen speaker. The main challenge of ZSM-TTS is to increase the overall speaker similarity for unseen speakers. One of the most successful speaker conditioning methods for flow-based multi-speaker text-to-speech (TTS) models is to utilize the functions which predict the scale and bias parameters of the affine coupling layers according to the given speaker embedding vector. In this letter, we improve on the previous speaker conditioning method by introducing a speaker-normalized affine coupling (SNAC) layer which allows for unseen speaker speech synthesis in a zero-shot manner leveraging a normalization-based conditioning technique. The newly designed coupling layer explicitly normalizes the input by the mean and standard deviation parameters predicted from a speaker embedding vector while training, enabling an explicit inverse process of denormalizing for a new speaker embedding at inference. The proposed conditioning scheme yields the state-of-the-art performance in terms of the speech quality and speaker similarity in a ZSM-TTS setting.