I'm afraid that's a common myth. For starters, there isn't really an objective measure of how conservative or old-fashioned a language is. But even subjectively, there's not really a reason to perceive American English as more old-fashioned.
All dialects randomly conserve and innovate on features. And when a dialect is split off by, in those case, geography, they will start to conserve and innovate different features. And it's true that American English conserved rhoticity which was lost in most of England, but it did innovate e.g. /æ/-raising where British English conserved the original pronunciation. But that's just two of many more features, and neither geographical region clearly conserved more than the other.
All dialects randomly conserve and innovate on features. And when a dialect is split off by, in those case, geography, they will start to conserve and innovate different features. And it's true that American English conserved rhoticity which was lost in most of England, but it did innovate e.g. /æ/-raising where British English conserved the original pronunciation. But that's just two of many more features, and neither geographical region clearly conserved more than the other.