This paper deals with a class of supercritical quasilinear Schrödinger equations

$$-\mathrm{\Delta }u+V(x)u+\kappa \mathrm{\Delta }(\sqrt{1+u^2})\frac{u}{2\sqrt{1+u^2}}=\lambda f(u),x\in {\mathbb{R}}^N,$$

where $\kappa \ge 2,N\ge 3,\lambda >0.$ We suppose that the nonlinearity $f(t):\mathbb{R}\to \mathbb{R}$ is continuous and only superlinear in a neighbourhood of $t=0.$ By using a change of variable and the variational methods, we obtain the existence of positive solutions for the above problem.