In grade school, my English teachers insisted upon three things: two spaces after a period, all sentences must have a subject, verb and object, NEVER start a sentence with "but." These were laws. Breaking them meant deductions on your paper, often stamped atop the document in red with an accusing circle highlighting the offending sentence.
As a thriller writer I can say with gusto that all "laws were meant to be broken."
Authors aim to have a conversation with readers through their point-of-view characters. When I'm in a character's head, I endeavor to write the way they would think--with a bit more editing. Few people think in proper sentences. Declarative statements and words are thrown into the mix of subject-verb-object complete thoughts. Actual dialogue trails off...
Here's a sentence from Ulysses: "God, isn't he dreadful? he said frankly. A ponderous Saxon. He thinks you're not a gentleman."
"A ponderous saxon" is all a subject. But, it's a good one.
And, that prior sentence started with "but." Take that fourth grade.