Do you power the servo motor with the Arduino 5V pin ? That 5V pin does not supply enough current for a servo motor. Test you sketch without the servo motor connected.
You use pin 3, set it as OUTPUT and make it HIGH and LOW. That should be okay.
Perhaps the contacts of the breadboard are bad. Or the relay coil requires too much current. Or the wrong transistor is used.
Could you tell which relay and which transistor you use, and could you make a photo, so we can see the wiring ?