Electronic processes leading to dissociative ionization of polyatomic molecules in strong laser fields are investigated experimentally, theoretically, and numerically. Using time-of-flight ion mass spectroscopy, we study the dependence of fragmentation on laser intensity for a series of related molecules and report regular trends in this dependence on the size, symmetry, and electronic structure of a molecule. Based on these data, we develop a model of dissociative ionization of polyatomic molecules in intense laser fields. The model is built on three elements: (i) nonadiabatic population transfer from the ground electronic state to the excited-state manifold via a doorway (charge-transfer) transition; (ii) exponential enhancement of this transition by collective dynamic polarization of all electrons, and (iii) sequential energy deposition in both neutral molecules and resulting molecular ions. The sequential nonadiabatic excitation is accelerated by a counterintuitive increase of a large molecule's polarizability following its ionization. The generic theory of sequential nonadiabatic excitation forms a basis for quantitative description of various nonlinear processes in polyatomic molecules and ions in strong laser fields.