I gather you want the browser to display on your Windows computer, not the Pi, so you don't need an X server running on PiCore, but you do want one running on Windows (that seems to be what MobaXterm is).
You've got Xorg running on the Pi, but that shouldn't harm anything, it's just not required. If MobaXterm works like OpenSSH then it should automatically export the DISPLAY environment variable so that programs are opened on the Windows computer's X server instead of Xorg on the Pi. Maybe try connecting without Xorg running on the Pi in case something's not working right with how MobaXterm does that.
Also if you have another computer running Linux and Xorg, try using "ssh -X" to do the same thing from that instead of the Windows computer, just to see whether the problem is specific to MobaXterm.