The following explanation is authored by Kevin L. O'Brien, firstname.lastname@example.org:
There are two views of how evolution works. The first is called anagenetics. This is where one type gradually evolves into another type along a single line. The second is cladogenetics. This is where a population splits and each part evolves separately along a different line from the other. In the first case one species literally replaces another as the parent species evolves into the daughter species. In the second case the new and old species live side by side until one or both becomes extinct. The new species does not replace the old one, and sometimes the old species can outlive the new species. It should be noted that nothing in these definitions assumes that the new species drives the old species to extinction.
[At times a lack of sufficient data may make it difficult to decide whether the process was anagenetic or cladogenetic; see p. 300, Lucy, the Beginnings of Humankind, by Johanson and Edy. By the same token, lack of sufficient data does not necessarily rule out either option.]
Creationists ignore cladogenetics and assume that all of evolution takes place anagenetically. Since they also have a distorted view of natural selection as survival of the fittest due to direct competition, they assume that the daughter species- in order to establish itself permanently - must wipe out the parent species, replace it entirely. As such, the creationists will claim that if a creature supposed to be extinct actually turns up alive in the modern world it obviously wasn't replaced by whatever species was supposed to have evolved in its place. Therefore anagenetics cannot be true. Therefore evolution cannot be true. It's a twisted, distorted argument that relies on the ignorance of the audience to be believed, but it crops up every now and then. Just recently when evidence was found suggesting that Homo erectus, which was supposed to have gone extinct 100,000 years ago, might in fact have survived until 25,000 years ago, Marvin Lubenow used it as evidence that evolution was false, since Home erectus obviously was not wiped out by whatever species was supposed to replace it. This argument is especially ludicrous in the case of the plesiosaur because they were made extinct, not by a competing new species, but by whatever general disaster claimed the dinosaurs. In any event, this argument is obviously wrong, because just as children do not kill their parents when they reach sexual maturity, a new species can live side by side with an old species for millions of years, until one finally goes extinct.
The age of the earth argument is based on the idea that it would be extremely difficult for a living fossil to survive 60 million years of continuously changing habitats and global environment, so if plesiosaurs are still alive it is more likely that the earth is only a few thousand years old. This assumes that plesiosaurs were created at the same time as everything else, survived the flood, but became less numerous as the world's oceans steadily grew colder. The few that are left could account for sea and lake monster reports, but otherwise they will continue to decline in numbers until they finally go extinct. This argument is also false because some living fossils have survived for a hundred million years at least, if not more. The coelocanth and the horseshoe crab are two examples. So it would not be difficult for plesiosaurs to survive 60 million years, so long as their niche was not eliminated by changing habitat environment.