Answer
If Earth and the moon had the same mass, indeed they would rotate around a common point exactly halfway between them.
However, the Earth is so much more massive than the moon (by a factor of about 80) that the Earth and Moon rotate about a point, known as the barycenter of the Earth-moon system, that lies within the Earth.
The barycenter is about 4700 km (about 3/4 of an Earth radius) from the Earth’s center.
