In order to prevail in a lawsuit, you must be able to show that the other party was negligent and due to their negligence, they caused you damage.  Generally, when you make a left turn into the path of an oncoming vehicle, you are liable for the accident.  It is possible in this instance both drivers share fault for the accident.  You for making the left turn into the path of an oncoming vehicle and the other driver for speeding and using their turn signal when they did not intend to turn.  Your question did not state whether or not you were injured in the accident or if you simply had damage to your vehicle.  Either way, your best course of action would be to make an insurance claim with either your own automobile insurance carrier, or the other driver's insurance carrier, or both.