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Photonics provide an efficient way to implement quantum walks, the quantum analog of classical random walks, which demonstrate rich physics with potential applications. However, most photonic quantum walks do not involve photon interactions, which limits their potential to explore strongly correlated many-body physics of light. We propose a strongly interacting discrete-time photonic quantum walk using a network of single atom beamsplitters. We calculate output statistics of the quantum walk for the case of two photons, which reveals the strongly correlated transport of photons. Particularly, the walk can exhibit either bosonlike or fermionlike statistics which is tunable by postselecting the two-photon detection time interval. Also, the walk can sort different types of two-photon bound states into distinct pairs of output ports under certain conditions. These unique phenomena show that our quantum walk is an intriguing platform to explore strongly correlated quantum many-body states of light. Finally, we propose an experimental realization based on time-multiplexed synthetic dimensions.