I'm sorry but you have to try and draw a cleared schematic. because this one is extremely hard to follow,
- The component values are not always present
- Try avoid crossing unconnected wires as much as possible
- Name important nets (HV+, HV-, signal grounds, inputs, etc)
- show in some way what is mounted or not (that could just mean crossing out stuff that is not soldered)
- Space things out, no designator should be covered by another component
- The bridge legs MUST be drawn according to the usual convention with the positive terminal at the top and the negative at the bottom, and the output in the middle
- Everything must flow from left to right, that means not only the gate signals but also the power, so on the far right there should be the Half bridge output, not the input
Also most importantly, the circuit is not complete, you should show everything related to the power, as I don't know what all the unmarked lines going out of the shown section do.
Now, with all of that out of the way there are few things that I can't understand, how is the high side gate driver powered or is it bootstrap?
what are those 50 ohm resistor, RZH and RZL
anyway the purple trace on the scope is very wrong, the gate should raise to the gate drive voltage, with a speed set by the gate resistor, and then it should stay there until the other transistor is turned on, apart from a small dead time (1-2% of the period or less) just enough to be sure to avoid cross conduction, in your case the gate rises to ~15V and then the gate driver suddenly switches off
also a 500 kHz switching frequency for that mosfet is way too high (that means by at least a factor 20) as the transistors will cook to death in a matter of milliseconds. To switch 600V at 10A at those frequency you need to look at wide bangap devices, which are way more expensive, fragile (especially GaN HEMT that die for just about anything, including looking at them wrong)
Also keep in mind that even if you manage to switch those at such a frequency (at no more than 10 or 20 V of course), how will you control the resulting converter, a microcontroller is out of question as you will be hard pressed to close even a trivial current loop at 80-100 kHz (and that is with the hand optimized code ecc), that leaves you with FPGA, and frankly that is a big step up in complexity