L298N H-Bridge burned up

Thank you.

Can a Seeeduino handle that much current? I thought 1k6 would serve OK as switch.

Did you forget the ~0.7volt drop of the BE junction of the transistor.
There will only be 3.3-0.7= 2.6volt across the resistor.
So 1K3 will theoretically put 2mA into the base.
But... the MCU pin will also have some loss, so 1k could be needed for that 2mA.
Just use 1k2 and call it a day.

Typically a speed-up resistor would be directly from base to emitter and be low in value to expedite clearing stored charge during switch-off - often a similar value to the base resistor.

Based on figure 11 It looks like I reached V(sat) using a 1K6 resistor