Work in a large saturated model, $\mathcal M.$ Things like $x$ and $y$ are not assumed to be singletons. They may be tuples.
First, I want to be careful about just what the question says. To say that this relation is type definable would mean that as a subset of $\mathcal M^2,$ the set of points satisfying the relation is the intersection of definable sets in $\mathcal M^2.$
Lets use the following characterization of forking - everything is over the empty set, but I want to assume that the types are stationary. $a$ forks with $b$ if $tp(a/b)$ represents a new formula. In a stable theory, types are definable, that is, fix a formula $\psi(x,y)$ and consider the set of parameters $c$ so that $b \models \psi (x,c).$ Definability of types says that this is a definable set. In fact, it is a boolean combination of instances of $\psi .$
So, for each new formula that $a$ might represent over $b,$ we can simply tell, definably. Intersecting over the set of all potential new formulas which $a$ might represent gives a type-definable subset of $\mathcal M^2.$
Maybe someone else can comment: is there a way to get around stationarity in a simple manner? Maybe one should just do things over the algebraic closure of the empty set and somehow go back down? Am I just being silly? (I am used to working over models, not over the empty set...)