Valley-layer coupling (VLC) two-dimensional (2D) materials define a distinct class of quantum systems in which valleys are related by crystal, rather than time-reversal (T) symmetry, enabling gate-controlled valley-contrasted layer polarization. Here we extend this concept to excitons. Using TiSiCo as a prototype VLC material, we show that a perpendicular electric field E_⊥ controls exciton- dispersion anisotropy and thereby generates unconventional linear transverse exciton transport in both monolayer and twisted bilayer structures. In the monolayer, this response is characterized by anisotropy-induced transverse conductivities, arising from the antisymmetric combination of diagonal elements in the conductivity tensor and strongly tunable by E_⊥. In the twisted bilayer, symmetry additionally permits transverse responses from the symmetric part of the conductivity tensor. Since intralayer excitons in different layers are connected by the Förster coupling, these symmetric and antisymmetric responses coexist and compete with the recently proposed T-even layer Hall and Nernst exciton counterflow. This interplay is highly tunable by twisted angle, temperature, and the direction of the in-plane driving force, providing a route to disentangle distinct transverse exciton transport signals experimentally. Our results establish E_⊥ as a powerful knob for controlling exciton transport via VLC and identify VLC 2D materials as a promising platform for engineered excitonic phenomena.