In the study of relativistic jets one of the key open questions is their interaction with the environment. Here we study the initial evolution of both electron-proton ({e}^{--}-{p}⁺) and electron-positron (e^±) relativistic jets, focusing on their lateral interaction with ambient plasma. We follow the evolution of toroidal magnetic fields generated by both the kinetic Kelvin-Helmholtz and Mushroom instabilities. For an {e}^{--}-{p}⁺ jet, the induced magnetic field collimates the jet and electrons are perpendicularly accelerated. As the instabilities saturate and subsequently weaken, the magnetic polarity switches from clockwise to counterclockwise in the middle of the jet. For an e^± jet, we find strong mixing of electrons and positrons with the ambient plasma, resulting in the creation of a bow shock. The merging of current filaments generates density inhomogeneities that initiate a forward shock. Strong jet-ambient plasma mixing prevents a full development of the jet (on the scale studied), revealing evidence for both jet collimation and particle acceleration in the forming bow shock. Differences in the magnetic field structure generated by {e}^{--}-{p}⁺ and e^± jets may contribute to the polarization properties of the observed emission in AGN jets and gamma-ray bursts.