We study the magnetic flux landscape in YBa$_2$Cu$_3$O$_{7-x}$ thin films as a two dimensional rough surface. The vortex density in the superconductor forms a self-affine structure in both space and time. This is characterized by a roughness exponent $α = 0.76(3)$ and a growth exponent $β = 0.57(6)$. This is due to the structure and distribution of flux avalanches in the self-organized critical state, which is formed in the superconductor. We also discuss our results in the context of other roughening systems in the presence of quenched disorder. Comment: 13 pages, 7 figures, accepted for publication in Physica A