jmc

Suppose Billy starts at point $A$, turns at point $B$, and ends at point $D$, as shown below. If Billy turns $60^{\circ }$ northward and walks six miles, then we can draw a $30-60-90$ triangle whose hypotenuse is $6$ miles (triangle $BCD$ below).

It follows that Billy traveled $6/2=3$ miles eastward during these $6$ miles, and that he traveled $3\cdot \sqrt{3}$ miles northward during these $6$ miles. In total, Billy traveled $4+3=7$ miles eastward and $3\sqrt{3}$ miles northward. By the Pythagorean Theorem, the distance from his starting point is $\sqrt{(7{)}^2+(3\sqrt{3}{)}^2}=\sqrt{49+27}=\sqrt{76}=\boxed{2\sqrt{19}}.$