In terms of power consumption, yes. But in terms of electrical noise that crashes the Arduino, this may be a very different situation.Hopefully if I can get the sketch uploaded it should work. The motors actually came from this car kit so it should all work together
The car kit probably had more sophisticated circuitry driving the motors, and probably had buffering capacitors, potentially some current limiting, etc. which would reduce the impact of the motors on the Arduino itself.
Your relay kit would provide even better isolation and protection if you used separate power supplies, but when trying to run it all off of one battery, it's probably creating a worse situation for the Arduino than the car kit motor drivers did.