Aims. We aim to clarify the origin of the multiple jet features emanating from the binary protostar SVS 13A (=VLA4A/VLA4B). Methods. We used the Plateau de Bure Interferometer to map at 0.3–0.8′′ (~70–190 au) dust emission at 1.4 mm, CO(2–1), SiO(5–4), SO(6₅–5₄). Revised proper motions for VLA4A/4B and jet wiggling models are computed to clarify their respective contribution. Results. VLA4A shows compact dust emission suggestive of a disk <50 au, and is the hot corino source, while CO/SiO/SO counterparts to the small-scale H₂ jet originate from VLA4B and reveal the jet variable velocity structure. This jet exhibits ≃ 3″ wiggling consistent with orbital motion around a yet undetected ≃20–30 au companion to VLA4B, or jet precession. Jet wiggling combined with velocity variability can explain the large apparent angular momentum in CO bullets. We also uncover a synchronicity between CO jet bullets and knots in the HH7–11 chain demonstrating that they trace two distinct jets. Their ≃300 yr twin outburst period may be triggered by close perihelion approach of VLA4A in an eccentric orbit around VLA4B. A third jet is tentatively seen at PA ≃ 0°. Conclusions. SVS13 A harbors at least 2 and possibly 3 distinct jet sources. The CO and HH7-11 jets are launched from quasi-coplanar disks, separated by 20–70 au. Their synchronous major events every 300 yr favor external triggering by close binary interactions, a scenario also invoked for FU Or outbursts.