Why does a bromine atom have a much smaller radius than a potassium atom, even though a br atom has 16 more electrons than does a k atom?

Good question!

As you can see in the periodic table, bromine and potassium lies in the same period. It is true that atomic size depends on the number of electron shells but since K and Br have the same period, they would have same number of electron shells. But why the size of the atom is still different between K and Br? The secret lies in the electrons.
Yea you are on the right track of thinking that bromine have more electrons than potassium. The reason why bromine atom have a smaller radius is very much related to the forces of attraction.
Think about it...
Which forces of attraction is stronger?
1) between 1e and 1p (valence e of K)
2) between 7e and 7p (valence e of Br)

The answer is 2.

I am not sure whether you can get the idea that actually the more electrons you have, the stronger the attraction force.
Basically the more number of electrons an atom have, the stronger the forces of attraction and thus, the more compact or smaller the atom.

Noble gases is far right