You're confusing the number of reflections (4) with the increase in size from the cyclic rotation group of size 4 to the dihedral group of size 8 (a factor of 2

You're confusing the number of reflections (4) with the increase in size from the cyclic rotation group of size 4 to the dihedral group of size 8 (a factor of 2). Note that any reflection can be decomposed into a fixed reflection followed by a rotation.