Context. The 51 Eridani system harbors a complex architecture with its primary star forming a hierarchical system with the binary GJ 3305AB at a projected separation of 2000 au, a giant planet orbiting the primary star at 13 au, and a low-mass debris disk around the primary star with possible cold and warm components inferred from the spectral energy distribution. Aims. We aim to better constrain the orbital parameters of the known giant planet. Methods. We monitored the system over three years from 2015 to 2018 with the Spectro-Polarimetric High-contrast Exoplanet REsearch (SPHERE) instrument at the Very Large Telescope (VLT). Results. We measure an orbital motion for the planet of ~130 mas with a slightly decreasing separation (~10 mas) and find a hint of curvature. This potential curvature is further supported at 3σ significance when including literature Gemini Planet Imager (GPI) astrometry corrected for calibration systematics. Fits of the SPHERE and GPI data using three complementary approaches provide broadly similar results. The data suggest an orbital period of 32₋₉⁺¹⁷ yr (i.e., 12₋₂⁺⁴ au in semi-major axis), an inclination of 133₋₇⁺¹⁴ deg, an eccentricity of 0.45_−0.15^+0.10, and an argument of periastron passage of 87₋₃₀⁺³⁴ deg [mod 180°]. The time at periastron passage and the longitude of node exhibit bimodal distributions because we do not yet detect whether the planet is accelerating or decelerating along its orbit. Given the inclinations of the orbit and of the stellar rotation axis (134–144°), we infer alignment or misalignment within 18° for the star–planet spin-orbit. Further astrometric monitoring in the next 3–4 yr is required to confirm at a higher significance the curvature in the motion of the planet, determine if the planet is accelerating or decelerating on its orbit, and further constrain its orbital parameters and the star–planet spin-orbit.