The Institute of Nuclear Physics SB RAS in cooperation with the Russian Federal Nuclear Center VNIITF performs a series of researches aimed at acquiring a relative electron beam with energy up to 20 MeV, current up to 2 kA, and duration up to 200 ns at normalized emittance ca. 1000 π∙mm∙rad in a linear induction accelerator (LIA). In order to generate electron beams with such parameters we require thorough investigation of all main sources of perturbation of electron beam trajectory caused by different instabilities that occur during the transport and acceleration of a high currency beam in the accelerating structure of LIA. For the experimental series on measuring the dynamics of transverse oscillation of a beam, we applied a set of fast current transformers which are used for registration of beam current and mode fields caused by this beam in the structure. These measurements were performed for the electron beam with the energy of 8.5 MeV and current of 1 kA going through the structure at different modes of focusing magnetic fields size in LIA. As a result, we registered oscillation of the electromagnetic field of normal modes in the accelerating modules of LIA, as well as we determined the dependence of the oscillation amplitude of these modes’ EM field on the number of accelerating module. This dependence was compared with the result of modeling of development dynamics of transverse instability in LIA that was performed using the created program system. This allowed us to determine the size of the increment of transverse instability of a relativistic electron beam under the given experimental conditions. Based on the acquired results, we made the analysis of possibility to use the beam generated in ILA as a driver for the FEL generator of coherent impulses of THz radiation within the frequency range of 0.3–1.2 THz with a sub-gigawatt level of power.