It is still challenging to achieve agility and trajectory control for untethered soft robots on an insect scale given their low mechanical impedance and compact structures. In this study, fast translational movements and swift turning motions are demonstrated on a power autonomous soft robot with a piezoelectric-thin-film-actuated body and electrostatic turning footpads. A high relative running speed of 2.5 body length per second compared with existing untethered robots is realized on a 24-mm-long untethered prototype integrated with power source, control, and wireless communication modules. An arc-shaped leg structure is adopted to self-regulate the frication forces on different footpads during turning by an inclination-induced redistribution of the payload gravity on legs and footpads. The trajectory maneuverability is demonstrated by navigating a 380 mg robot prototype with an 1810 mg payload to pass through a 58-cm-long S-shaped path with wireless control in 43.4 s. Due to the flexibility of the all-polymer body structure, the robustness of the untethered robot to large strain is demonstrated when compressed by 91 times the weight of the robot. A maximum travel distance of 58.6 m is achieved for the robot equipped with a 40 mA·h lithium battery, corresponding to the cost of transport of 261. This work provides a feasible solution to achieve high agility and advance the practicability of untethered soft robots on an insect scale.