The influence of self-generated magnetic and electric fields on the transport of relativistic electrons in dense plasmas was studied using a particle-in-cell simulation. For conditions relevant to the fast ignitor, the laser-driven relativistic electrons may have significant energy spread along or perpendicular to the propagation direction of the beams. The effect of electron energy spread on the growth rate, the occurrence threshold of both the Weibel-like filamentation instability and the electrostatic two-stream instability and the competition between them were investigated. The Weibel instability results in the formation of a magnetic channel, which may collimate inward fast electrons without significant deviation, and the excitation of a longitudinal electric field due to two-stream instability is destructive to the stability of the magnetic channel. The generation of relativistic electrons by the interaction of a high-intensity laser beam at the vacuum–dense plasma boundary and propagation of the electrons in the dense plasma were studied using a three-dimensional particle-in-cell code. It is shown that the electron velocity spread owing to transverse collective heating saturates the magnetic field and the longitudinal electrostatic field may play a dominant role in limiting the stable propagation of fast electrons into over-dense plasmas.