We study the delocalization in interacting disordered quasi-1D and 2D systems. The results are strikingly similar to the 1D case, with slow, subdiffusive dynamics featuring power-law decay. From the freezing of this decay we infer the critical disorder strength as a function of length and width of the sample. In the quasi-1D case with a fixed width, the critical disorder strength has a finite large-length limit which rapidly grows with increasing width. In the 2D case the critical disorder grows with increasing system size. The results are consistent with the avalanche picture of the many-body localization transition.