Context. The interaction of the solar-wind plasma with a magnetized planet generates a bow-shaped shock ahead of the wind. Over recent decades, near-Earth spacecraft observations have provided insights into the physics of the bow shock, and the findings suggest that solar-wind intrinsic turbulence influences the bow shock dynamics. On the other hand, theoretical studies, primarily based on global numerical simulations, have not yet investigated the global three-dimensional (3D) interaction between a turbulent solar wind and a planetary magnetosphere. This paper addresses this gap for the first time by presenting an investigation of the global dynamics of this interaction that provides new perspectives on the underlying physical processes. Aims. We use the newly developed numerical code MENURA to examine how the turbulent nature of the solar wind influences the 3D structure and dynamics of magnetized planetary environments, such as those of Mercury, Earth, and magnetized Earth-like exoplanets. Methods. We used the hybrid particle-in-cell code MENURA to conduct 3D simulations of the turbulent solar wind and its interaction with an Earth-like magnetized planet through global numerical simulations of the magnetosphere and its surroundings. MENURA runs in parallel on graphics processing units, enabling efficient and self-consistent modeling of turbulence. Results. By comparison with a case in which the solar wind is laminar, we show that solar-wind turbulence globally influences the shape and dynamics of the bow shock, the magnetosheath structures, and the ion foreshock dynamics. Also, a turbulent solar wind disrupts the coherence of foreshock fluctuations, induces large fluctuations on the quasi-perpendicular surface of the bow shock, facilitates the formation of bubble-like structures near the nose of the bow shock, and modifies the properties of the magnetosheath region. Conclusions. The turbulent nature of the solar wind impacts the 3D shape and dynamics of the bow shock, magnetosheath, and ion foreshock region. This influence should be taken into account when studying solar-wind-planet interactions in both observations and simulations. We discuss the relevance of our findings for current and future missions launched into the heliosphere.