High-energy proton beams have extensive and important applications. Traditional proton accelerators are bulky and costly. The development of high-power laser pulse technology provides a new proton acceleration scheme based on the interaction between laser and plasma, which has the advantage of miniaturization. Furthermore, compared with traditional proton accelerators, the proton acceleration gradient by high-power laser pulses can be increased by three orders of magnitude. The high brightness, narrow pulse width, and good directionality proton beams can be generated in theory within a very small effective size, and they are suitable for fields such as nuclear physics and particle physics, ion beam fast ignition, medical treatment, proton beam detection, etc. In order to realize laser proton acceleration, a large amount of research with different target configurations and acceleration mechanisms has been reported on proton acceleration driven by ultrashort and high-power lasers. However, due to the limitation of laser intensity, the energy of proton beam driven by a single-beam laser is difficult to improve to meet the needs of medical applications. In this paper, a new method of driving proton acceleration by multiple ultrashort high-power lasers with grazing incidence on both sides of the microstrip target is proposed. With two-beams driving settings, a proton beam with energy divergence of about 3% and energy of about 165 MeV can be obtained. The simulation results with two-dimensional particle-in-cell show that the laser extracts a large number of collimated high-energy electron charges on both sides of the solid target and injects them into the rear of the target. A longitudinal bunching field is built at the rear of the target to drive proton acceleration and bunch to form a quasi-monoenergetic high-energy proton beam. The research also shows that the proton beam with energy divergence of about 2% and energy of about 250 MeV can be obtained by using four grazing ultrashort high-power lasers on both sides of the microstrip target. The mechanism of multi-laser beams driving proton acceleration provides a new idea for the energy enhancement of the proton beam, and the quasi-monoenergetic high-energy proton beam is expected to be applied in the field of medical treatment.