  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.
