It takes time for a driver who has been convicted of a crime to be able to get cheap car insurance again. However, it is possible if the driver does good things and stops doing the bad things that led to the conviction.
If you get a DUI or DWI conviction...
...make sure you do everything the court needs you to do. This could mean that you have to take a defensive driving course or a similar course and pass it. You might also have to start going to meetings for Alcoholics Anonymous or Narcotics Anonymous. Your judge may even order you to go through a programme to help you stop using drugs and alcohol. No matter what your sentence is, make sure you finish it. One day, you might be able to get cheap car insurance, but until then, you need to show that you are responsible.
If you are caught breaking the rules of the road...
...it's time to start taking care of your driving record. Most traffic violations don't get you arrested or even cause your licence to be taken away, but they do add points to your driving record. If you want to lower the number of points on your driving record, ask about taking a defensive driving course. Aside from that, you just have to wait for the points to go away on their own, which could take three years or more.
If you are found guilty of causing an accident on the road...
...again, you must meet all of the court's requirements. Make sure that your current car insurance covers all injuries and damage to property. Take as many defensive driving courses and other courses like them as you can. Ask what you can do to reduce the number of points on your driving record, and be ready to wait a few years for them to go away.
Getting cheap car insurance after a conviction isn't easy or quick, but if you follow the steps and have patience, you'll get cheap car insurance again one day.