| The divine comedy|
Many[Who?] believe in human-dinosaur coexistence, i.e., that since humans have been around since the beginning of time, they necessarily coexisted with nonavian dinosaurs. This is even shoehorned into a belief that dinosaurs are mentioned in the Bible (e.g., the Behemoth from Job, dragons). The open disagreements with creationists fall mostly on how the dinosaurs were kept on Noah's Ark—how the Ark would have been large enough, how they died out afterwards, and so on. A further sub-group have also claimed that dinosaurs aren't extinct and that there were recent findings of dinosaurs in the remote places of Earth. An example is Kent Hovind who believes that today's lizards are dinosaurs that have lost the genes for largeness.[note 1] The more insane believe that dinosaurs are Satan's own creation. According to "Tobacco Truthers", smoking killed all the dinosaurs, as revealed by Gary Larson in "The Far Side." Carl Everett, a former MLB player, also said dinosaurs did not exist and that their fossils are fakes (were you seriously expecting someone who was on the Red Sox to be intelligent or something?). 
Why this is bollocks
There's nothing particularly special about dinosaurs in this case. There are countless extinct species, some coming after the nonavian dinosaurs went extinct, many coming before that. Insects, for example, date back to around 400 million years ago; a number of the extinct prehistoric species are far larger than modern equivalents due to higher atmospheric oxygen concentrations, as insects don't possess respiratory systems that allow them to grow to an indefinite size. Without the higher oxygen concentration, such species would not survive - but at the same time, such higher oxygen concentrations would be less conducive to the existence of other species. Human coexistence with these species is certainly more problematic for creationists, but dinosaurs are in the popular imagination for being large land animals unlike any that currently exist, so they remain as a great image for creation museums to use.
The particular problem with human-dinosaur coexistence is the same problem presented by all flood geology. Namely, that in the fossil record there is absolutely no trace of human remains found in the layers where dinosaur fossils are found. There are no modern animals mixed in with dinosaur remains; there are no dinosaur remains mixed in with more modern animals. In fact, the fossil record is remarkably well-ordered for being supposedly, according to creationists, put down by a devastating and chaotic flood.
In the real world (or, "Why it matters")
In June of 2012, journalists discovered that publishers of fundamentalist Christian textbooks would begin issuing textbooks claiming the Loch Ness Monster is a real live plesiosaur[note 2]. These texts would be used by Christian schools and homeschoolers to promote denialism of the theory of evolution. One critic believed that they would be used in at least thirteen American states.
It should be noted that not all people who believe in human-dinosaur coexistence are necessarily creationists, but simply ignorant and believed what they saw on The Flintstones. The cure for this is simple: education.
For the pedants in the audience
These days, biologists regard modern birds as extant dinosaurs. Although the popular image is that dinosaurs were like lizards, there are a lot of differences between large reptiles seen today (e.g., the Komodo dragon, crocodiles, and so on) and dinosaurs. The legs of all extant lizards and crocodiles, for example, sprawl out to the sides while all dinosaur species have their legs extending directly below the body. Upright bipedal dinosaurs (theropods), then, have more in common with the bone structures of large birds than of modern reptiles. There's significant evidence that dinosaurs had lungs closer in operation to modern birds for improved efficiency, especially in larger dinosaurs for which mammalian and modern reptilian lungs simply wouldn't have worked. The embryonic and evolutionary origins of scales and feathers are also practically identical, and many non-avian dinosaurs had feathers not too dissimilar to those of modern birds.
In fact, dinosaurs are defined[note 3] as everything descended from the most recent common ancestor of Triceratops and its descendants including ... Passer domesticus. The house sparrow. There's your human-coexisting dinosaur and who is actually mentioned in the Bible.
- The Mysterious Origins of Man - an entire documentary dedicated to proving man and dinos co-existed
- Jurassic Park - Less wrong
- Paluxy River tracks
- TheLivingDinosaur - YouTuber debunking creationism who takes his nom de plume (pun intended) from this issue
- Unicorn - much more real according to (some versions of) the bible
- Ignore that this would be evolution happening and run with it, m'kay?
- Ignore that this is a salt-water creature that they claim are living in a fresh water environment. If they were right, that would probably require evolution to happen
- There is, of course, fiddling around the edges of this. See dinosaur.
- Why do people laugh at creationists? (part 7): In which Hovind claims that a Triceratops was a Jackson's Chameleon
- Bible Answers Live Question Archive:Are dinosaurs Satan's animals?
- McKin, Claire. "Loch Ness monster cited by US schools as evidence that evolution is myth", Edinburgh Scotsman website, published and accessed 26 June 2012.
- Take a gander at this thread at Godlike Productions even they think it is stupid. Mostly.
- Natural History Museum - What were dinosaurs?
- Pharyngula - Dinosaur lungs