Modeling dynamic fracture in solids, particularly under high-velocity impact and large deformations, presents significant challenges for mesh-based methods due to severe mesh distortion, presents significant challenges for mesh-based methods due to severe mesh distortion. This work proposes a robust computational framework integrating a fourth-order phase-field fracture model within Eulerian Smoothed Particle Hydrodynamics (ESPH). While second-order phase-field models are widely utilized, they often suffer from slower convergence and broader damage profiles. The proposed fourth-order formulation introduces higher-order spatial derivatives, providing superior regularity of the crack surface field and sharper damage gradients. This enhancement is critical for accurately capturing complex crack topologies, such as branching and fragmentation, without the need for ad-hoc visibility criteria or explicit crack tracking required in traditional SPH fracture methods. To address the computational demands of dynamic problems, we employ a hyperbolic formulation of the phase-field evolution equation. Unlike the standard parabolic form, the hyperbolic governing equation facilitates efficient explicit time integration, making it highly suitable for wave propagation and impact dynamics. Numerical validation is performed through dynamic benchmarks, including dynamic crack branching plates and asymmetric bending tests. The results demonstrate that the coupled fourth-order phase-field ESPH framework accurately predicts crack tip velocities and branching patterns consistent with experimental observations. This approach may offer a promising avenue for simulating catastrophic failure in protective structures under extreme loading conditions.