Charge transport in conjugated polymer semiconductors has traditionally been thought to be limited to a low-mobility regime by pronounced energetic disorder. Much progress has recently been made in advancing carrier mobilities in field-effect transistors through developing low-disorder conjugated polymers. However, in diodes these polymers have to date not shown much improved mobilities, presumably reflecting the fact that in diodes lower carrier concentrations are available to fill up residual tail states in the density of states. Here, we show that the bulk charge transport in low-disorder polymers is limited by water-induced trap states and that their concentration can be dramatically reduced through incorporating small molecular additives into the polymer film. Upon incorporation of the additives we achieve space-charge limited current characteristics that resemble molecular single crystals such as rubrene with high, trap-free SCLC mobilities up to 0.2 cm2/Vs and a width of the residual tail state distribution comparable to kBT.