We investigate charge injection and transport in organic field-effect transistors fabricated by using poly(2,5-bis(3-tetradecylthiophene-2-yl) thieno[3,2-b]thiophene) (pBTTT-C14) or poly(3-hexylthiophene) (P3HT) as the active polymer layer. We show that in high mobility devices where the channel resistances are low compared to the contact resistances, the device performance can be dominated by the metal/organic semiconductor (OSC) contacts. However, in sets of devices where the channel resistance is dominant over the contacts (usually the lower mobility devices), we see pronounced field dependence in the saturation regime mobilities consistent with a Poole-Frenkel model of charge transport within the channel. The field-dependent mobility in short-channel devices produces nonlinear output current-voltage characteristics which can be modeled consistently in the Poole-Frenkel framework.