Nope. The bible in no way is unable to explain the existence of dinosaurs. Evolution is not the only answer.
Since fossils exist, we can reasonably believe that dinosaurs exist. That's elementary. Now on to the more exciting controversial segment: how did they came to be?
Some believe in abiogenesis, which basically theorizes that living organisms came about from non-living things. Through chemical interactions, proteins for living things came about. How this happens in a sterile and hostile environment nobody can say for sure. From abiogenesis, a single-celled organism would have evolved into more complex organisms, such as dinosaurs. (Evolution has its own problems, which I would not into here).
Some believe in creationism, which theorizes that God created all living organisms based on their kind. This includes dinosaurs. It makes more sense, but you must assume the existence of God, which is what secular scientists and many laymen refuse to do.
Genesis 1:20-24
"20 And God said, "Let the water teem with living creatures, and let birds fly above the earth across the expanse of the sky." 21 So God created the great creatures of the sea and every living and moving thing with which the water teems, according to their kinds, and every winged bird according to its kind. And God saw that it was good. 22 God blessed them and said, "Be fruitful and increase in number and fill the water in the seas, and let the birds increase on the earth." 23 And there was evening, and there was morningthe fifth day. 24 And God said, "Let the land produce living creatures according to their kinds: livestock, creatures that move along the ground, and wild animals, each according to its kind." And it was so. 25 God made the wild animals according to their kinds, the livestock according to their kinds, and all the creatures that move along the ground according to their kinds. And God saw that it was good."
I'm sure you're very familiar with this passage. Yes you're right, dinosaurs are big reptiles (and small reptiles, and whatever in between) and they constitute 'creatures that move along the ground'. Dinosaurs in the sea and dinosaurs that fly were created one day earlier.
There isn't a direct reference to dinosaurs, just like there isn't a direct reference to the horse that David rode on, or the name of his mother's best friend. It's one of those details that may not have been necessary to be in the bible. Nonetheless, the poster before me did mention leviathans.
Job 3:8
May those who curse days curse that day, those who are ready to rouse Leviathan.
Job 41:1
"Can you pull in the leviathan with a fishhook or tie down his tongue with a rope?
Psalm 74:14
It was you who crushed the heads of Leviathan and gave him as food to the creatures of the desert.
Psalm 104:26
There the ships go to and fro, and the leviathan, which you formed to frolic there.
Isaiah 27:1
[ Deliverance of Israel ] In that day, the LORD will punish with his sword, his fierce, great and powerful sword, Leviathan the gliding serpent, Leviathan the coiling serpent; he will slay the monster of the sea.
Now if you read job you'll know that leviathan resembles a dragon like creature, which can symbolically refer to Satan. Nonetheless in Job, the description of leviathan seems to imply that he was an amphibious dinosaur.
Job 40:15
"Look at the behemoth, which I made along with you and which feeds on grass like an ox.
The behemoth likewise sounds like a sauropod.
These are vague references to dinosaurs, and like I said, there are no direct references, but the 1st chapter in genesis does include the creation of dinosaurs.
