Water displays anomalous fast diffusion in narrow carbon nanotubes (CNTs), a behavior that has been reproduced in both experimental and simulation studies. However, little is reported on the effect of bulk water-CNT interfaces, which is critical to exploiting the fast transport of water across narrow carbon nanotubes in actual applications. Using molecular dynamics simulations, we investigate here the effect of such interfaces on the transport of water across arm-chair CNTs of different diameters. Our results demonstrate that diffusion of water is significantly retarded in narrow CNTs due to bulk regions near the pore entrance. The slowdown of dynamics can be attributed to the presence of large energy barriers at bulk waterCNT interfaces. The presence of such intense barriers at the bulkCNT interface arises due to the entropy contrast between the bulk and confined regions, with water molecules undergoing high translational and rotational entropy gain on entering from the bulk to the CNT interior. The intensity of such energy barriers decreases with increase in CNT diameter. These results are very important for emerging technological applications of CNTs and other nanoscale materials, such as in nanofluidics, water purification, nanofiltration, and desalination, as well as for biological transport processes.