I've powered it with 12V battery. Driver's IN4 pin is linked with Arduino pin which is always set to HIGH. Arduino is power by computer USB port.
The problem is that on the motor output chanel A I have 1.6V output. Is this because Arduino and controller does not have common ground? If it is so, can I just link Arduino's ground with drivers ground leaving usb power?
You must have a GND connection to the Arduino to complete the circuit for the PWM signal.
A photo of a pencil drawing is better than an image with coloured lines on it that does not identify the names of the pins. The names of the pins you are not using is also helpful.
What pin on the L298 are you using? I think you need to activate at least one of the H-bridge pins and enable (plus GND).