We study the global bifurcation diagram of the positive solutions to the problem{Δu+λf(u)=0inB,u=0on∂B, where B is the unit ball in RN with N≥3. Under general supercritical growth conditions on f(u), we show that an unbounded bifurcation curve has no turning point, which indicates the existence of the singular extremal solution. In particular, our theory can be applied to the super-exponential cases of f(u), and we exhibit that a bifurcation curve for Δu+λf(u)=0 has the same qualitative property as a classical Gel'fand problem Δu+λeu=0 for N≥3 except N=10. Main technical tools are intrinsic transformations for semilinear elliptic equations and ODE techniques.