We provide numerical evidence for slow dynamics of the susceptible-infected-susceptible model evolving on finite-size random networks with power-law degree distributions. Extensive simulations were done by averaging the activity density over many realizations of networks. We investigated the effects of outliers in both highly fluctuating (natural cutoff) and nonfluctuating (hard cutoff) most connected vertices. Logarithmic and power-law decays in time were found for natural and hard cutoffs, respectively. This happens in extended regions of the control parameter space λ 1 < λ < λ 2 , suggesting Griffiths effects, induced by the topological inhomogeneities. Optimal fluctuation theory considering sample-to-sample fluctuations of the pseudothresholds is presented to explain the observed slow dynamics. A quasistationary analysis shows that response functions remain bounded at λ 2 . We argue these to be signals of a smeared transition. However, in the thermodynamic limit the Griffiths effects loose their relevancy and have a conventional critical point at λ c = 0. Since many real networks are composed by heterogeneous and weakly connected modules, the slow dynamics found in our analysis of independent and finite networks can play an important role for the deeper understanding of such systems.