We performed extensive Monte Carlo simulations of site-disordered two-dimensional Ising systems along the critical line T(c)(p) in the concentration range of magnetic sites between p = 0.6 and p = 1.0 (p(c) = 0.59). Magnetic and caloric properties were studied as well as the cumulants of the magnetization distribution. We found that the disorder induces a crossover phenomenon in the experimental temperature range which leads to a strong, concentration-dependent increase of gamma(eff)(gamma(max) = 2.16 for p = 0.7) in the temperature range 10(-2) < (T - T(c))/T(c) < 10(0). Near T(c), gamma(eff) asymptotically approaches gamma = 1.75, independent of the concentration. We propose a model of weakly coupled compact clusters of spins to describe this crossover phenomenon.