Turning left requires more power to overcome the increased induced drag caused by the increase in angle of attack on the rear of the rotor disc.
To turn left, the AOA must increase at the rear of the disc. Due to precession, this change is felt on the right side of the disc and turns the helicopter left. In forward flight, the front of the disc receives airflow more horizontally, while the rear of the disc has more induced flow. As a result, the AOA changes to turn left requires a significant increase in the AOA. This increase caused more induced drag on the rotor disc. To maintain the same rotor speed, more power is required to overcome the increased drag. The opposite is true when turning right. A right turn changes the AOA on the front of the disc, which is operating with less induced flow. Also, the reduction in the AOA on the rear of the disc reduces induced drag and less power is required.
The impact of this change is more significant at higher speeds and higher gross weight. In high performance turbine helicopters, abrupt changes from right to left can place significant stress on the aircraft systems.
Like most items in a helicopter, there are many different forces working with and/or against each other. In a turn, the increased g forces from the turn also speed up the disc as there is more weight.
Fundamentals of Flight pg. 1-141