The superlinear \((p> 1)\) non-autonomous Sturm-Liouville problem \(- u''+ k(x) u^p= \lambda u\) in \((0, 1)\), \(u(0)= u(1)= 0\) is studied from the standpoint of \(L^2\)-theory. For the solution \((\lambda, u_\lambda)\) \((\lambda> \pi^2, u_\lambda> 0)\), the bifurcation diagram in \(L^2\times \mathbb{R}\) is investigated. That is, \(\lambda\) and \(u_\lambda\) are regarded as a function of \(\alpha= |u_\lambda|_{L^2}\) and asymptotic formulas of \(\lambda= \lambda(\alpha)\) and \(u_\lambda= u(\alpha, u)\) as \(\alpha\to \infty\) with optimal remainder estimate are established.