> There are an infinite number of functions that can create a particular bit sequence, and each bit sequence can be mapped to a single string, so there must be more functions than strings.
I don't think this logic works. For example, there are an infinite number of "rational numbers" of the form np/p where n, p ∈ ℤ, and each of these can be mapped to a single integer (namely, n). But there are not more these "rational numbers" than integers.
When you have an injection from A to B, you only know that B is at least as equinumerous as A. Maybe their cardinality is the same, and maybe the cardinality of B is strictly bigger.
For your example, we can construct an injection from rational numbers to integers, as well as an injection from integers to rational numbers. Therefore, by the Schröder–Bernstein theorem they are equinumerous.
I don't think this logic works. For example, there are an infinite number of "rational numbers" of the form np/p where n, p ∈ ℤ, and each of these can be mapped to a single integer (namely, n). But there are not more these "rational numbers" than integers.