Now, what your saying here is the survival of the fittest. But if the animals that are weaker are always dying out quickly, how do they have millions of years to evolve into something that will survive?
Not everything that was around a long time ago is around now. Things evolve and things go extinct, it's how life works. Also a better version of a creature doesn't mean that that creature is the only one to be able to survive there. So over millions of years small changes to benefit species end up helping creatures survive a lot.
And the "weaker" animals don't always die out quickly. In fact just because they were around longer doesn't always mean they are weaker. If you have one a car, then you take another car of the same type and glue a box to the top of it it doesn't really mean either one is much much worse than the other, aside from one being more aesthetically pleasing.
There is also (forgot the name, "separated evolution" I will call it.) separated evolution, in which one species is split into 2 groups because of a natural formation. I.e. a group of squirrels are split up because a river forms between the 2. The 2 squirrel species cannot mate with each other because of the river, so over time both types of squirrels adapt differently to their surroundings, resulting in 2 different species of squirrel.
EDIT: Also, it doesn't have to be where the new, better suited species has to whipe out the other species. But the traits it has generally get passed on more because of their increased survival rate, eventually making the older ones more rare.