This paper considers the following parabolic-elliptic chemotaxis-growth system with nonlinear diffusion $\left\{ \begin{array}{l}{u_t} = \nabla (D(u)\nabla u) - \nabla (\chi {u^q}\nabla v) + \mu u(1 - {u^\alpha }),\;\;\;\;\;\;\;\;& x \in \Omega ,{\mkern 1mu} {\mkern 1mu} t > 0,\\0 = \Delta v - v + {u^\gamma },\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&x \in \Omega ,{\mkern 1mu} {\mkern 1mu} t > 0\end{array} \right.$ under homogeneous Neumann boundary conditions for some constants $q≥ 1$, $α>0$ and $γ≥ 1$, where $D(u)≥ c_D u^{m-1}$$(m≥ 1)$ for all $u>0$ and $D(u)>0$ for all $u≥ 0$, and $Ω\subset\mathbb{R}^N$ $(N≥ 1)$ is a bounded domain with smooth boundary. It is shown that when $ m>q+γ-\frac{2}{N}, \, \, \mathbf{or}$$ α>q+γ-1, \, \, \mathbf{or}$$α = q+γ-1\, \, {\rm{and}}\, \, μ>μ^*$, where $ {\mu ^*} = \left\{ \begin{array}{l}\begin{array}{*{20}{l}}{\frac{{(\alpha + 1 - m)N - 2}}{{(\alpha + 1 - m)N + 2(\alpha - \gamma )}}\chi ,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{if}}~~{\mkern 1mu} {\mkern 1mu} m \le q + \gamma - \frac{2}{N},}\end{array}\\0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{if}}~~{\mkern 1mu} {\mkern 1mu} m > q + \gamma - \frac{2}{N},\end{array} \right.$ then the above system possesses a global bounded classical solution for any sufficiently smooth initial data. The results improve the results by Wang et al. (J. Differential Equations 256 (2014)) and generalize the results of Zheng (J. Differential Equations 259 (2015)) and Galakhov et al. (J. Differential Equations 261 (2016)).