What you are trying to do is going to almost impossible for a smartphone, at least not without a fair bit of editing work. For starters, make sure your lens is cleaned. You'd be surprised how much stuff like body oils build up on it, will make things hazy.
Unfortunately, the lenses themselves are low quality compared to dedicated cameras. Trying to take photos of direct lights like this will almost always have some sort of blooming like you see. Increasing the shutter speed and lowering ISO will help, to a point. Of course that also means a darker photo overall and you'll lose all the details in the rest of the photo without additional lighting.
Here's what I'd suggest. Get a smartphone tripod mount and take two photos from the exact same location. In one, set the manual mode controls to get the headlights the best you can. In the other, turn the lights off and set the camera to expose everything else properly. If needed, use a timer to countdown while you hop in the car and press the brake pedal to light up the rear while it takes the photo. This may take several attempts to get either photo correctly exposed (One for the headlights, one for the rest), but get your best one of each and make a composite of them in your photo editor of choice.