Gate signal problem on my flyback
I don't have any answers about distorted gate drive signals but I'll list down a few mistakes I see:

For \$\mathrm{P_{OUT} = 24W}\$ and \$\mathrm{f_{SW} = 500kHz}\$, the primary inductance will be much higher than \$\mathrm{15\mu H}\$ even for DCM. Nevertheless, let's see what that \$\mathrm{15\mu H}\$ causes:

If you force the converter to output 24 Watts then the controller chip will increase the on-time to maintain the required energy which will be stored by the primary inductance (its magnetic field, actually) then transferred to secondary (Remember \$\mathrm{V_{IN} = L_p\ i_{ppk}/t_{on}}\$, and \$\mathrm{E_p=0.5\ L_p\ i_{ppk}^2}\$), but the duty cycle is limited to ~%48 by the controller chip, as seen in the oscillogram.

For \$\mathrm{L_p = 15\mu H}\$ and \$\mathrm{t_{on} = 1\mu s}\$ (i.e. limited duty-cycle), the peak current will be ~7.3A and the RMS current will be 3A. That's why you see ~1A input current at 110VDC input.


So, maybe the distortion is caused by the duty-cycle limitation. Increase the primary inductance to something around \$\mathrm{100-120\mu H}\$ and see what happens.
NOTE: The transformer of a flyback converter must have an air gap because most of the energy is stored by the gap itself.
By the way, since you didn't share the full schematic, I'm assuming that the loop is properly closed.
PS: Say hello to Oguz Bey and Selman Bey for me :)