  We consider the statistical properties of arrival times and balls on
first-passage percolation (FPP) $2D$ square lattices with strong disorder in
the link-times. A previous work showed a crossover in the weak disorder regime,
between Gaussian and Kardar-Parisi-Zhang (KPZ) universality, with the crossover
length decreasing as the noise amplitude grows. On the other hand, this work
presents a very different behavior in the strong-disorder regime. A new
crossover length appears below which the model is described by bond-percolation
universality class. This characteristic length scale grows with the noise
amplitude and diverges at the infinite-disorder limit. We provide a thorough
characterization of the bond-percolation phase, reproducing its associated
critical exponents through a careful scaling analysis of the balls. This is
carried out through to a continuous mapping of the FPP passage time into the
occupation probability of the bond-percolation problem. Moreover, the crossover
length can be explained merely in terms of properties of the link-time
distribution. The interplay between the new characteristic length and the
correlation length intrinsic to bond-percolation determines the crossover
between the initial percolation-like growth and theasymptotic KPZ scaling.
