We consider a family of growth models defined using conformal maps in which the local growth rate is determined by $|Φ_n'|^{-η}$, where $Φ_n$ is the aggregate map for $n$ particles. We establish a scaling limit result in which strong feedback in the growth rule leads to one-dimensional limits in the form of straight slits. More precisely, we exhibit a phase transition in the ancestral structure of the growing clusters: for $η>1$, aggregating particles attach to their immediate predecessors with high probability, while for $η<1$ almost surely this does not happen.