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Clarifying the interplay of interactions and disorder is fundamental to the understanding of many quantum systems, including superfluid helium in porous media¹, granular and thin-film superconductors^{2, 3, 4, 5} and light propagating in disordered media^{6, 7, 8}. One central aspect for bosonic systems^{9, 10, 11} is the competition between disorder, which tends to localize particles, and weak repulsive interactions, which instead have a delocalizing effect. As the required degree of independent control of disorder and of interactions is not easily achievable in most available physical systems, a systematic experimental investigation of this competition has not so far been possible. Here we use a degenerate Bose gas with tunable repulsive interactions in a quasiperiodic lattice potential to study this interplay in detail. We characterize the entire delocalization crossover through the study of the average local shape of the wavefunction, the spatial correlations and the phase coherence. Three different regimes are identified and compared with theoretical expectations^{12, 13, 14, 15, 16, 17}: an exponentially localized Anderson glass and the formation of locally coherent fragments as well as a coherent, extended state. These results provide insight into the role of weak repulsive interactions in disordered bosonic systems, but our approach should also enable investigations of disordered systems with interactions in the strongly correlated regime^{18, 19, 20}.