Dinosaurs and other extinct animals can easily fit into a recent creation framework. The book of Job is certainly one of the earliest books of the Bible to be written down. In this book, two creatures are mentioned that could easily be so-called prehistoric animals. Behemoth (Job 40:15-24) is often identified with the elephant by Bible scholars. However, that will only work if we do not believe the text to be accurate. He moves his tail like a cedar tree (Job 40:17). This indicates a massive tail, not the small tail of an elephant. There are other reasons as well. The description sounds more like one of the dinosaurs than anything else.
Then there is leviathan (Job 41:1-34). Leviathan is a type of the devil (Isaiah 27:1), but God describes him to Job as a living sea creature. And what a creature he is! No man can catch him. He breathes fire out of his mouth. He cannot be turned to flight by spears or arrows. The description is of a creature like none known to exist today.
There is no reason to doubt that these animals existed in the time of Adam and beyond. They may have even been on Noah's ark--probably as young (and much smaller) animals. However, it is likely that the change in climate that occurred after the flood made it difficult for them to survive and they were soon extinct. Dinosaur prints have been found alongside of human footprints in some places, so it is probable that they existed at the same time.
Since I do not have absolute statements from the Bible, I cannot say that this is the only possible solution. However, according to what information the Bible gives us, I believe that this is the most likely solution.