In this paper, we study an intelligent reflecting surface (IRS)-aided radar-communication (Radcom) system, where the IRS is leveraged to help Radcom base station (BS) transmit the joint of communication signals and radar signals for serving communication users and tracking targets simultaneously. The objective of this paper is to minimize the total transmit power at the Radcom BS by jointly optimizing the active beamformers, including communication beamformers and radar beamformers, at the Radcom BS and the phase shifts at the IRS, subject to the minimum signal-to-interference-plus-noise ratio (SINR) required by communication users, the minimum SINR required by the radar, and the cross-correlation pattern design. In particular, we consider two cases, namely, case I and case II, based on the presence or absence of the radar cross-correlation design and the interference introduced by the IRS on the Radcom BS. For case I where the cross-correlation design and the interference are not considered, we prove that the dedicated radar signals are not needed, which significantly reduces implementation complexity and simplifies algorithm design. Then, a penalty-based algorithm is proposed to solve the resulting non-convex optimization problem. Whereas for case II considering the cross-correlation design and the interference, we unveil that the dedicated radar signals are needed in general to enhance the system performance. Since the resulting optimization problem is more challenging to solve as compared with the case I, the semidefinite relaxation (SDR) based alternating optimization (AO) algorithm is proposed. Particularly, instead of relying on the Gaussian randomization technique to obtain an approximate solution by reconstructing rank-one solution, the tightness is achieved by our proposed reconstruction strategy. Simulation results demonstrate the effectiveness of proposed algorithms and also show the superiority of the proposed scheme over various benchmark schemes.