  It has been shown in a previous work that torsional Alfv\'en waves can drive
turbulence in nonuniform coronal loops with a purely axial magnetic field. Here
we explore the role of the magnetic twist. We model a coronal loop as a
transversely nonuniform straight flux tube, anchored in the photosphere, and
embedded in a uniform coronal environment. We consider that the magnetic field
is twisted and control the strength of magnetic twist by a free parameter of
the model. We excite the longitudinally fundamental mode of standing torsional
Alfv\'en waves, whose temporal evolution is obtained by means of
high-resolution three-dimensional ideal magnetohydrodynamic numerical
simulations. We find that phase mixing of torsional Alfv\'en waves creates
velocity shear in the direction perpendicular to the magnetic field lines. The
velocity shear eventually triggers the Kelvin-Helmholtz instability (KHi). In
weakly twisted magnetic tubes, the KHi is able to grow nonlinearly and,
subsequently, turbulence is driven in the coronal loop in a similar manner as
in the untwisted case. Provided that magnetic twist remains weak, the effect of
magnetic twist is to delay the onset of the KHi and to slow down the
development of turbulence. In contrast, magnetic tension can suppress the
nonlinear growth of the KHi when magnetic twist is strong enough, even if the
KHi has locally been excited by the phase-mixing shear. Thus, turbulence is not
generated in strongly twisted loops
