In this paper, we study the classification and evolution of bifurcation curves of semipositone problem with Minkowski-curvature operator \begin{equation*} \left\{ \begin{array}{l} -\left( u^{\prime }/\sqrt{1-{u^{\prime }}^{2}}\right) ^{\prime }=\lambda f(u),\text{ \ in }\left( -L,L\right) , \\ u(-L)=u(L)=0, \end{array} \right. \end{equation*} where $\lambda ,L>0$, $f\in C^{2}(0,\infty )$, and the following conditions (H$_{1}$)--(H$_{2}$) hold: \begin{itemize} \item[(H$_{1}$)] there exists $\beta >0$ such that $(\beta -u)f(u)0$ and $u\neq \beta .$ \item[(H$_{2}$)] Let $F(u)\equiv \int_{0}^{u}f(t)dt$. Then $F:[0,\infty )\longrightarrow \mathbb{R}$ is continuous and differentiable for $u>0$. And there exists $\eta >0$ such that $(\eta -u)F(u)0$ and $u\neq \eta $. \end{itemize} \noindent Notice that we allow $f(0^{+})=-\infty $. In particular, we further obtain the exact shapes of the bifurcation curve as $f$ is convex or concave. Finally, we apply these results in several problems.