infinity - There's a real between any two rationals, a rational between any two reals, but more reals than rationals?

The following statements are all true:

• Between any two rational numbers, there is a real number (for example, their average).


• Between any two real numbers, there is a rational number (see this proof of that fact, for example).


• There are strictly more real numbers than rational numbers.

While I accept each of these as true, the third statement seems troubling in light of the first two. It seems like there should be some way to find a bijection between reals and rationals given the first two properties.

I understand that in-between each pair of rationals there are infinitely many reals (in fact, I think there's $2^{\aleph_0}$ of them), but given that this is true it seems like there should also be in turn a large number of rationals between all of those reals.

Is there a good conceptual or mathematical justification for why the third statement is tue given that the first two are as well?

Thanks! This has been bothering me for quite some time.