I'm not aware of any convincing evidence that dinosaurs (as we commonly understand that term) and humans co-existed. I suspect that if there were any such evidence, it would be very famous and very greatly discussed. Below, I've appended two links. One makes a case for the co-existence of humans and dinosaurs; the other cites many scientists who refute such claims, including the following: Bird, 1939; Neufeld, 1975; Zuidema, 1979 and 1981; Weber, 1981; Godfrey, 1981; Slaughter, in Kirsch, 1982; Langston, 1983.
The Reign of Dinosaurs ended about 65 million years ago, if we can trust the fossil and geologic record. That same record suggests that humans, as we would know them to be, didn't exist until 4 to 6 million years ago. Clearly what the authors of the Bible were recording were not dinosaurs, but as others have suggested, other species that have since gone extinct.
I have to agree with speamerfam. The Bible was not written (or so I am thinking--given that all answers will not be answered until death) to make any statement on evolution. The Bible is read, and interpreted, by those who wish to either follow (or dispute) the Word of God. It is not scientific.
The Bible neither proves nor disproves evolution. It serves an entirely different purpose, to tell stories about the relationship between man and God. Whether one believes those stories is a matter of faith, not of science.
There is some evidence that humans did indeed live among wooly mammoths and mastodons, which existed long after dinosaurs disappeared, according to commonly agreed upon scientific timelines. In fact, many scientists believe that humans helped push these animals to extinction. Mammoths and mastodons were huge and probably incredibly impressive to see, and we know that they did eat grass and they preferred swampy and damp areas. Is it possible that the verse references such a creature?
As I recall, the Leviathan is usually thought by biblical scholars to be some sort of whale.
The thing is, if dinosaurs lived among people, then practically every kind of science is wrong. It's not just a dispute over evolution. You then have to dispute things like geology and physics (because the dating of fossils is done by looking at the rates at which one kind of carbon turns into another).
Couldn't it just as well be that the Bible is God's word but put into ideas that could be understood by scientifically ignorant people living thousands of years ago?