Reliability as well as efficiency of transmissions is of a paramount issue for the safety critical services of VANET due to the urgency and/or significance of the messages. To this end,various retransmission mechanisms have been studied for reliable broadcast message delivery in VANETs. Most of the existing approaches, though, rely on periodic beacons for retransmission decisions and intrinsically require high control overheads. Especially, in urban environments where the traffic density tends to be high, minimizing control overheads is important not only for the efficiency but also for the reliability of user message delivery. In this paper, we propose an efficient broadcast message delivery mechanism, which does not rely on the periodic beacon exchanges. Based on the observation that the broadcast message delivery could be particularly difficult at the intersections due to the signal diffraction and attenuation caused by the obstacles such as buildings around the corners of the intersection, a novel mechanism to make efficient and effective retransmissions across an intersection is proposed. Simulation results show that the proposed mechanism can provide similar or better reliability while incurring prominently less overheads than the existing approaches.