According to the fossil record, the plesiosaur died out 66 million years ago, but the fossil record is incomplete, so we cannot use that argument to say plesiosaurs did not survive beyond that date.
Plesiosaur remains were discovered as early as 1605 and the creature was named in 1821. It is presumed it was a cold-blooded creature. Because of its long neck and humped back, it conforms in some respects to descriptions of lacustrine and marine monsters. Also, creatures washed up on the shore sometimes look like plesiosaurs.
With regard to the creatures washed onto the shoreline, they generally turn out to be the harmless basking shark. When one of these starts to decompose, sections of it rot away, leaving a remnant that resembles the outline of a plesiosaur. The Zuiyo-Maru monster, fished out of the sea in 1977, proved to be a decaying basking shark.
Sometimnes monsters, including him of Loch Ness, are described as beings with a hump and a head on the end of a slender neck, held upright. From this we can infer that such monsters are certainly not plesiosaurs, as it has been shown that it was quite impossible for them to raise their heads.
Whatever sea serpents and the Loch Ness Monster may be, we can rule out plesiosaurs.