Too little current to trigger a relay?
You are using pin 13, but you set pin 10 as an OUTPUT.

pinMode(10, OUTPUT);

Try changing that to 

pinMode(pinOut, OUTPUT);

PS by not setting pin 13 to OUTPUT, digitalWrite(pinOut, HIGH); enables the internal pull-up, instead of "connecting" it to 5V. The internal resistors are around 35kOhm. So pulling a bit of current through them will result in a lower voltage at the pin.