Time delay has always been one of the main factors affecting the application performance of neural network (NN) systems, and dynamic performance research of NNs with time delays has been the focus of many scholars in recent years. This article enquires into the exponentially synchronous problem of switched delayed NNs with time delay in the leakage term. Adopting an unusual form from a common switched system, the switching modes of the switched delayed NNs system in this article are dependent on time delays. In the first place, the master, slave, and error NNs models are reconstructed into the switched form by introducing the switched delay idea. Then with the help of the admissible edge-dependent average dwell time (AED-ADT) method and delay-dependent switching adjustment indicators, a novel set of generalized delay-mode-dependent multiple Lyapunov-Krasovskii functionals (MLKFs) is built for analyzing the cases where a state-feedback controller exists and does not exist in the model, and where parts of LKFs may increase during the period when the corresponding subsystems are activated. For these cases, several effective exponential synchronization criteria and switching laws are presented accordingly. At last, the verification of the theoretical results is shown through a few examples.