A one-way {em street} of width M is modeled as a set of M parallel one-dimensional TASEPs. The intersection of two perpendicular streets is a square lattice of size M times M. We consider hard core particles entering each street with an injection probability alpha. On the intersection square the hard core exclusion creates a many-body problem of strongly interacting TASEPs and we study the collective dynamics that arises. We construct an efficient algorithm that allows for the simulation of streets of infinite length, which have sharply defined critical jamming points. The algorithm employs the `frozen shuffle update, in which the randomly arriving particles have fully deterministic bulk dynamics. High precision simulations for street widths up to M=24 show that when alpha increases, there occur jamming transitions at a sequence of M critical values alphaM,M < alphaM,M-1 < ... < alphaM,1. As M grows, the principal transition point alphaM,M decreases roughly as sim 1/(log M) in the range of M values studied. We show that a suitable order parameter is provided by a reflection coefficient associated with the particle current in each TASEP.
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