How to fix Arduino restarting when two servos are powered by an external power?
You will need to fix this on your own but I can suggest some things that you will likely need to investigate.


Evaluate what exactly causes the AVR to get reset. It is highly likely that long wires and current surges as loads are applied is causing either the GND or the VCC to the AVR to be compromised and initiating a brown out reset in the MCU.
Driving inductive loads without proper design may be putting high voltage spikes into your design that create havoc with proper software execution and/or risking serious damage to the silicon components. Make sure you take steps to eliminating such with clamp diodes, filter caps or zener diodes.
You will have to check your exact MCU data sheet but many parts have a reset reason register that your software can read to see what particular thing initiated the reset.
Lastly the most important thing is to make sure you have the proper tools for debugging problems like this. It is very common that you should have access to an oscilloscope so you can find the problems and learn how to fix them. The best way to learn how to design in a way that you avoid problems like this on the next project is to see first hand what went wrong and learn from it.