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Choice C is correct. The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. Therefore, $a^2+b^2=c^2$, where $a$ and $b$ are the lengths of the legs and $c$ is the length of the hypotenuse. For the given right triangle, the lengths of the legs are $4$ and $b$, and the length of the hypotenuse is $19$. Substituting $4$ for $a$ and $19$ for $c$ in the equation $a^2+b^2=c^2$ yields $4^2+b^2={19}^2$. Thus, the relationship between the side lengths of the given triangle is $4^2+b^2={19}^2$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.