The Zakharov-Kuznetsov equation (ZK) is a model for the propagation of waves in the context of plasma physics and can be viewed as a two-dimensional analogue of the celebrated Korteweg-de Vries equation (KdV). In this talk, we study the Cauchy problem associated with the $k$-generalized Zakharov-Kuznetsov equation (gZK) posed on $\mathbb{R} \times \mathbb{T}$, where $k \geq 2$ is an integer. We establish several new Strichartz-type estimates in the framework of Jean Bourgain's $X_{s,b}$ spaces, with the main contributions being an almost optimal linear $L^4$-estimate and a family of bilinear refinements of this bound. As a direct application, we prove multilinear $X_{s,b}$-estimates that lead to improved local well-posedness thresholds for gZK via a fixed-point iteration.