Arduino Nano + L239DNE + external power supply
I doubt you mean L239 ....but actually mean L293

Here's the datasheet

Since you are only using a 6 V DC supply for the motor, you most likely have too low a voltage across the motor for it to work properly. 
The output stages drop voltage and you can see from the datasheet you lose about 1.8 + 1.2 = 3 V for the upper and lower driver portion.



If you are using a 6 V motor I'd suggest you need at least a 9 V DC supply.

You could measure the voltage across your motor with a multimeter to verify this is the problem.