There is no English rule that says a sentence cannot begin with and, but, or, nor. I truly believe the idea that this is a rule starts in elementary school, where we all hooked a bunch of sentences together with “and.” That well-meaning teacher said, “Don’t start a sentence with ‘and.'” For whatever reason, it stuck!
Though there is no rule, we seem reluctant to use one of these conjunctions at the beginning of a sentence. But sometimes we have to. For those times when a witness (or attorney or judge) just keep stringing sentences together with “and” or “or” or one of the other ones, call a halt. Start a new sentence.
The rule of thumb is no more than three complete sentences in one.
…We were driving northbound on Madison, and we turn left. And I saw a car in a driveway, and it started to move, and it didn’t stop. He backed right out in front of me, and I applied the brake. But it was too late, and I hit him.