When taming a high level megalosaurus, you need to put down right when night starts. It sounds counterproductive because it's stronger, but it allows for maximum starve time since you will need to put it down again the following night. If you don't do it this way, you may end up needing to put it down a third time, which could very likely be the death of it since it won't regain any health over that time.
Also, tame with a friend because the megalosaurus will pick you up into its jaws which makes it very hard to do solo without dying. Your tribe mate can attack the Dino and cause him to drop you.