High-energy proton beams have extensive and important applications. Traditional proton accelerators are bulky and costly. The high-power laser pulse technology provides a new proton acceleration scheme based on the interaction between laser and plasma, and has the advantage of miniaturization. Furthermore, comparing with traditional proton accelerators, the proton acceleration gradient by high-power laser pulses can be increased by three orders of magnitude. The proton beams with high brightness, narrow pulse width, and good directionality 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, and proton beam detection. In order to realize laser proton acceleration, a great many of researches of different target configurations and acceleration mechanisms have been reported on proton acceleration driven by ultrashort and high-power lasers. However, owing 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. A proton beam with an energy divergence of about 3% and energy of about 165 MeV can be obtained by using the two-beam driving setting. The results of two-dimensional particle-in-cell simulation show that a large number of collimated high-energy electron charges are extracted from both sides of the solid target by laser and injected into the back of the target. A longitudinal bunching field is established on the back of the target, which drives protons to accelerate and bunch to form a quasi-monoenergetic high-energy proton beam. The research also shows that the proton beam with an 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 to the field of medical treatment.