  We construct cobordisms of small genus between torus knots and use them to
determine the cobordism distance between torus knots of small braid index. In
fact, the cobordisms we construct arise as the intersection of a smooth
algebraic curve in $\mathbb{C}^2$ with the unit 4-ball from which a 4-ball of
smaller radius is removed. Connections to the realization problem of
$A_n$-singularities on algebraic plane curves and the adjacency problem for
plane curve singularities are discussed. To obstruct the existence of
cobordisms, we use Ozsv\'ath, Stipsicz, and Szab\'o's $\Upsilon$-invariant,
which we provide explicitly for torus knots of braid index 3 and 4.
