  Electron states with the spin-momentum-locked Dirac dispersion at the surface
of a three-dimensional (3D) topological insulator are known to lead to weak
antilocalization (WAL), i.e. low temperature and low-magnetic field quantum
interference-induced positive magnetoresistance (MR). In this work we report on
the MR measurements in (Bi,Sb)$_2$(Te,Se)$_3$ 3D topological insulator thin
films epitaxially grown on Si(111), demonstrating an anomalous WAL amplitude.
This anomalously high amplitude of WAL can not be explained by parabolic or
linear MR and indicates the existence of an additional, MR mechanism. Another
supporting observation is not linear in the classically weak magnetic field
Hall effect in the same films. The increase of the low-field Hall coefficient,
with respect to the higher-field value, reaches 10$\%$. We consistently explain
both transport features within a two-liquid model, where the mobility of one of
the components drops strongly in a weak magnetic field. We argue that this
dependence may arise from the Zeeman field induced gap opening mechanism.
