  The transverse structure of jets was studied via jet fragmentation transverse
momentum ($j_{\rm{T}}$) distributions, obtained using two-particle correlations
in proton-proton and proton-lead collisions, measured with the ALICE experiment
at the LHC. The highest transverse momentum particle in each event is used as
the trigger particle and the region $3 < p_{\rm{Tt}} < 15$ GeV/$c$ is explored
in this study. The measured distributions show a clear narrow Gaussian
component and a wide non-Gaussian one. Based on Pythia simulations, the narrow
component can be related to non-perturbative hadronization and the wide
component to quantum chromodynamical splitting. The width of the narrow
component shows a weak dependence on the transverse momentum of the trigger
particle, in agreement with the expectation of universality of the
hadronization process. On the other hand, the width of the wide component shows
a rising trend suggesting increased branching for higher transverse momentum.
The results obtained in pp collisions at $\sqrt{s}$ = 7 TeV and in p-Pb
collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV are compatible within
uncertainties and hence no significant cold nuclear matter effects are
observed. The results are compared to previous measurements from CCOR and
PHENIX as well as to Pythia 8 and Herwig 7 simulations.
