I understand that if we use a high-order Runge Kutta method like RK4 the rate of convergence of the error as the stepsize $h$ tends to $0$ should be of order $h^4$.
Does that necessarily imply that the global error (i.e. max norm of the difference between the exact and approximate solution) of say a fifth-order Runge Kutta method is lower than the error obtained with a fourth-order method? In other words, the error curve as a function of the stepsize of the fifth order method should be at least below the fourth order method, for all stepsizes?
No, it is not necessarily so. The error of an $m$'th order method is bounded by (and usually approximately equal to) $C h^m$ for some constant $C$. However, different methods will have different values of $C$. As a result, although for sufficiently small step size $h$ the higher-order method should beat the lower-order method, for any particular $h$ there are no guarantees.