For the trajectory tracking problem of under-actuated vehicle platoons under input delay, this study focuses on investigating the lateral and longitudinal stability of vehicle platoons during curve-following processes. An adaptive sliding mode control method for vehicle platoons based on prescribed performance and delay compensation is proposed. Based on path curvature and the state information of the preceding vehicle, an extended spacing strategy that accounts for input delay and curvature variations is designed, which can effectively enhance both lateral and longitudinal stability as well as safety performance. By transforming the under-actuated tracking model using a prescribed performance function, the tracking errors of the vehicle platoon are constrained within a fixed range. Furthermore, by integrating an adaptive compensation mechanism, an adaptive sliding mode control method for vehicle platoons that incorporates input delay and position tracking errors is proposed. This method ensures that the control system exhibits a fast convergence rate and higher steady-state accuracy. The fixed-time stability of the following vehicle and the lattice stability of the platoon are proven using the Lyapunov stability theory. Finally, the effectiveness and feasibility of the proposed method are validated through simulation experiments.