We introduce generalized $TAC$-contractive mappings in $b$-metric spaces and we prove some new fixed point results for this class of mappings. We provide examples in support of our results. Our results extend the results of [S. Chandok, K. Tas and A. H. Ansari, Some fixed point results for $TAC$-type contractive mappings, J. Function Spaces, Vol. 2016, Article ID 1907676, 6 pages] from the metric space setting to $b$-metric spaces and generalize a result of [D. Djorić, Common fixed point for generalized $(\psi,\varphi)$-weak contractions, Appl. Math. Lett. 22 (2009) 1896-1900].