Lyrics transcription of polyphonic music is challenging not only because the singing vocals are corrupted by the background music, but also because the background music and the singing style vary across music genres, such as pop, metal, and hip hop, which affects lyrics intelligibility of the song in different ways. In this work, we propose to transcribe the lyrics of polyphonic music using a novel genre-conditioned network. The proposed network adopts pre-trained model parameters, and incorporates the genre adapters between layers to capture different genre peculiarities for lyrics-genre pairs, thereby only requiring lightweight genre-specific parameters for training. Our experiments show that the proposed genre-conditioned network outperforms the existing lyrics transcription systems.