Solution-processed semiconductors such as conjugated polymers have great potential in large-area electronics. While extremely appealing due to their low-temperature and high-throughput deposition methods, their integration in high-performance circuits has been difficult. An important remaining challenge is the achievement of low-voltage circuit operation. The present study focuses on state-of-the-art polymer thin-film transistors based on poly(indacenodithiophene-benzothiadiazole) and shows that the general paradigm for low-voltage operation via an enhanced gate-to-channel capacitive coupling is unable to deliver high-performance device behavior. The order-of-magnitude longitudinal-field reduction demanded by low-voltage operation plays a fundamental role, enabling bulk trapping and leading to compromised contact properties. A trap-reduction technique based on small molecule additives, however, is capable of overcoming this effect, allowing low-voltage high-mobility operation. This approach is readily applicable to low-voltage circuit integration, as this work exemplifies by demonstrating high-performance analog differential amplifiers operating at a battery-compatible power supply voltage of 5 V with power dissipation of 11 µW, and attaining a voltage gain above 60 dB at a power supply voltage below 8 V. These findings constitute an important milestone in realizing low-voltage polymer transistors for solution-based analog electronics that meets performance and power-dissipation requirements for a range of battery-powered smart-sensing applications.