We report on the formation and evolution of two large-scale, synchrotron-emitting jets from the black hole candidate H1743-322 following its reactivation in 2003. In 2003 November, after the end of its 2003 outburst, we noticed, in observations with the Australia Telescope Compact Array, the presence of a new and variable radio source about 4."6 to the east of H1743-322 that was later found to move away from H1743-322. In 2004 February, we detected a radio source to the west of H1743-322, symmetrically placed relative to the eastern jet. In 2004, follow-up X-ray observations with Chandra led to the discovery of X-ray emission associated with the two radio sources. This likely indicates that we are witnessing the interaction of relativistic jets from H1743-322 with the interstellar medium, causing in situ particle acceleration. The spectral energy distribution of the jets during the decay phase is consistent with a classical synchrotron spectrum of a single electron distribution from radio up to X-rays, implying the production of very high energy (>10 TeV) particles in those jets. We discuss the jet kinematics, highlighting the presence of a significantly relativistic flow in H1743-322 almost a year after the ejection event.