I have tried reducing the low side gate resistance and increasing the high side gate resistance, but they don't seem to make any impact. I'd like to mention that the gate waveform is clean when there's no load, but when I connect a 20W bulb, I get the above waveforms. There are small spikes at the output as well, which align with the spikes at the gate.
It looks like the gate drive path to the bottom MOSFET is a bit long / has narrow traces, which could impact the gate drive.
I am not sure whether this is a problem. I found this post
https://www.eevblog.com/forum/projects/ltc3895-step-down-converter-burns-off/msg2098939/#msg2098939 where capt bullshot has attached a layout of a buck converter and the low side FET gate trace is long, thin and passes between the drain and source pads of the high side FET. However in my case, it passes above the switching node, but I don't think it would cause such a significant spike.
However, are you sure the FET is actually turning on? Are you seeing excessive heating / blowing FETs? The switching transition has fast transients that can couple into the scope probe. Are you using a spring ground lead? If you short the probe to the ground plane do you still see a large transient while switching?
This could be an issue, I am not using a spring ground lead, it's the conventional clip that is grounded at a point next to the input capacitor. The FETs get warm after a minute of operation with no heatsink attached at 20W load. None of the FETs has blown yet. The high side Vds and low side Vds appear to be clean. I concluded that dV/dt is the culprit and not the probe as it aligned with the high side turn on.
Are you probing with a low inductance spring clip probe? As was said, your gate drive traces are much too thin and long, and seem to go over switch node planes. Move the controller as close as possible to the FET gates, use a Kelvin connection to the FET sources, and increase trace width by maybe 5-10X. But first have to make sure the 5V spike is actually real and not a measurement artifact/noise.
I'll get a spring clip and redo the probing. I am planning to probe the input current and see if there's shoot through and will update this post.