Techniques are provided for avoiding an avalanche breakdown voltage across a synchronous rectification (SR) switch on the secondary side of an isolated switched-mode power converter operating in a low-power mode, e.g., a burst mode, during which a load of the power converter draws negligible current. This is accomplished via use of a two-level switch driver for controlling a power switch on the primary side of the power converter. The two-level switch driver is configured to source low current levels to a control terminal (e.g., gate) of the power switch during burst-mode operation. This low current reduces the slope of the rising edge of voltage pulses on the primary and secondary sides of the power converter which, in turn, limits the peak of the voltage ringing across the SR switch. By limiting the voltage in this manner, the SR switch avoids entering avalanche breakdown.