Yes, it is grammatically correct, but considered too informal for academic work. That is, "but" is a coordinate conjunction and joins two independent clauses. It is considered better style to use longer conjunctions, like "however"
It is also correct, btw, to start a sentence with "because" if there's another clause: Because it looked like rain, he took his umbrella. The problem is that students write just "because it looked like rain." So teachers tell them not to start sentences with "Because". But it's not grammatically wrong.