We study the class $\mathcal{L}$ of link-types that admit a $K_4$-minor-free diagram, i.e., they can be projected on the plane so that the resulting graph does not contain any subdivision of $K_{4}$. We prove that $\mathcal{L}$ is the closure of a subclass of torus links under the operation of connected sum. Using this structural result, we enumerate $\mathcal{L}$ (and subclasses of it), with respect to the minimum number of crossings or edges in a projection of $L\in \mathcal{L}$. Further, we obtain counting formulas and asymptotic estimates for the connected $K_4$-minor-free link-diagrams, minimal $K_4$-minor-free link-diagrams, and $K_4$-minor-free diagrams of the unknot.