I got a ultrasound guided injection on my plexus brachial area. After the injection I have more pain now & its on the left side too. Is this normal?

NO!! It is very difficult procedure do and the person did who did it on you obviously did not succeed! I know a great anesthesiologist in NYC but that doesn't help you. The migration of the pain to the other side may be another process - possibly a slipped disc in the neck? Either go back the same anesthesiologist or find another one probably at UCLA or UCSF.