YECs like the Creationist Museum, believe that dinosaurs did exsist. But they believed that dinosaurs.became extinct because of the Flood. Their theory is after the flood, the remaining dinosaurs that survived on the Ark eventually died out because there was a lack of sufficient food for other necessities for the dinosaurs. But this falls flat because if there was food for Noah, his family, and all the other paired animals right after the flood, then wouldn't the dinosaurs who went on the ark also had enough food for themselves as well?
Genesis 1:21 says that God created humongous sea creatures. Though I am studying for Marine Biology myself, I know that blue whales are the largest known marine animals and of all animals alive today. But, it is really possible that these creatures are the kind of creatures God also created in Genesis 1:21. Isaiah 51:9 also points out that Rahab, or humongous sea monsters like these below, we're pierced:
http://listverse.com/2013/01/12/10-terrifying-prehistoric-sea-monsters/
Furthermore, when the Earth was first created, it was without void. (Genesis 1:2) That was after Satan was kicked out of heaven. Since that void has lasted for billions of years, dinosaurs did exist millions of years ago in this Earth. During the void when Satan was on the Earth, there was chaos in creation for billions of years.