Arduino car on bluetooth remote appears to crash when motors started
it sounds like the motors are pulling down the battery voltage enough to reboot the Arduino; and the USB-port is able to supply enough current without as much voltage drop.

We'd need to see a schematic, but check that the power supplied to your Arduino and power supplied to the motors are completely independent; that the Arduino only supplies current to a switch or a relay in the motors' power circuit but not to the motors themselves.