It is true that the moon makes one rotation around the earth in 27.3 days. A lunar month is the number of days between two new moon days. After one new moon day the moon does arrive at the same position with respect to the fixed stars after 27.3 days.
But it is not aligned with the earth and the sun because during this period the earth has moved some distance around the sun, indeed by about 270. So the moon has to move some more to come to a position between the sun and the earth.
A simple calculation shows that this additional traverse would take about 2.2 days. Therefore the next new moon comes after 29.5 days and not 27.3-the lunar month is longer than the period of rotation of the moon by about Two days!