Intelligent reflecting/refracting surface (IRS) has recently emerged as a promising solution to reconfigure wireless propagation environment for enhancing the communication performance. In this paper, we study a new IRS-aided high-mobility communication system by employing the intelligent refracting surface with a high-speed vehicle to aid its passenger's communication with a remote base station (BS). Due to the environment's random scattering and vehicle's high mobility, a rapidly time-varying channel is typically resulted between the static BS and fast-moving IRS/user, which renders the channel estimation for IRS with a large number of elements more challenging. In order to reap the high IRS passive beamforming gain with low channel training overhead, we propose a new and efficient transmission protocol to achieve both IRS channel estimation and refraction optimization for data transmission. Specifically, by exploiting the quasi-static channel between the IRS and user both moving at the same high speed as well as the line-of-sight (LoS) dominant channel between the BS and IRS, the user first estimates the LoS component of the cascaded BS-IRS-user channel, based on which IRS passive refraction is designed to maximize the corresponding IRS-refracted channel gain. Then, the user estimates the resultant IRS-refracted channel as well as the non-IRS-refracted channel for setting an additional common phase shift at all IRS refracting elements so as to align these two channels for maximizing the overall channel gain for data transmission. Simulation results show significant performance improvement of the proposed design as compared to various benchmark schemes. The proposed on-vehicle IRS system is further compared with a baseline scheme of deploying fixed intelligent reflecting surfaces on the roadside to assist high-speed vehicular communications, which achieves significant rate improvement.