Many real networks present a bounded scale-free behavior with a connectivity cutoff due to physical constraints or a finite network size. We study epidemic dynamics in bounded scale-free networks with soft and hard connectivity cutoffs. The finite size effects introduced by the cutoff induce an epidemic threshold that approaches zero at increasing sizes. The induced epidemic threshold is very small even at a relatively small cutoff, showing that the neglection of connectivity fluctuations in bounded scale-free networks leads to a strong overestimation of the epidemic threshold. We provide the expression for the infection prevalence and discuss its finite size corrections. The present paper shows that the highly heterogeneous nature of scale-free networks does not allow the use of homogeneous approximations even for systems of a relatively small number of nodes.