Hi thanks for the great question!

The first thing to try is to finger the Em chord with your middle finger on the 2nd fret of the A string and your ring finger on the 2nd fret of the D string. That minimizes the finger movement in grabbing the C chord by using your ring finger on the 3rd fret of the A string, middle finger on the 2nd fret of the D string, and your index finger on the 1st fret of the B string.

Experiment with that fingering and see if that makes a difference in making the chord change quicker. If not, try some other fingerings and see which one feels the most natural and comfortable for you and stick with it.

Then, the most important thing you can do, is to repeatedly drill that chord change over and over at a MUCH LOWER TEMPO. This is critical -- slow down the tempo so that you can make the change in time cleanly and consistently. Once you find a slower tempo that you can make the change nice and smooth -- keep practicing at that tempo over and over. Repetition is the key.

The more practice you put in at a slow tempo, your fingers and muscles will start to adapt and you'll find making the change easier and easier.

Then start to increase the tempo in small doses and repeat the same process. The idea is to keep inching up the tempo, lots of repetition, and eventually you'll be making the change cleanly and up to tempo!

Hope this helps, let me know how it goes!
Mike