This article is devoted to addressing the cloud control of connected vehicles, specifically focusing on analyzing the effect of bi-directional communication-induced delays. To mitigate the adverse effects of such delays, a novel predictor-observer structured controller is proposed which compensate for both measurable output delays and unmeasurable, yet bounded, input delays simultaneously. The study begins by novelly constructing an equivalent delay-free inter-connected system model that incorporates the Predictor-Observer controller, considering certain delay boundaries and model uncertainties. Subsequently, a stability analysis is conducted to assess the system's robustness under these conditions. Next, the connected vehicle lateral control scenario is built which contain high-fidelity vehicle dynamic model. The results demonstrate the controller's ability to accurately predict the system states, even under time-varying bi-directional delays. Finally, the proposed method is deployed in a real connected vehicle lateral control system. Comparative tests with a conventional linear feedback controller showcase significantly improved control performance under dominant bi-directional delay conditions, affirming the superiority of the proposed method against the delay.