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A thin, narrow-bandgap semiconductor Bi$_2$O$_2$Te nanosheet is obtained via mechanical exfoliation and a Hall-bar device is fabricated from it on a heavily doped Si/SiO$_2$ substrate and studied at low temperatures. Gate transfer characteristic measurements show that the transport carriers in the nanosheet are of $n$-type. The carrier density, mobility, and mean free path in the nanosheet are determined by measurements of the Hall resistance and the longitudinal resistance of the Hall-bar device and it is found that the electron transport in the nanosheet is in a quasi-two-dimensional (2D), strongly disordered regime. Magnetotransport measurements for the device at magnetic fields applied perpendicular to the nanosheet plane show dominantly weak antilocalization (WAL) characteristics at low fields and a linear magnetoresistance (LMR) behavior at large fields. We attribute the WAL characteristics to strong spin-orbit interaction (SOI) and the LMR to the classical origin of strong disorder in the nanosheet. Low-field magnetoconductivity measurements are also performed and are analyzed based on the multi-channel Hikami-Larkin-Nagaoka theory with the LMR correction being taken into account. The phase coherence length, spin relaxation length, effective 2D conduction channel number and coefficient in the linear term due to the LMR in the nanosheet are extracted. It is found that the spin relaxation length in the Bi$_2$O$_2$Te nanosheet is several times smaller than it in its counterpart Bi$_2$O$_2$Se nanosheet and thus an ultra-strong SOI is present in the Bi$_2$O$_2$Te nanosheet. Our results reported in this study would greatly encourage further studies and applications of this emerging narrow-bandgap semiconductor 2D material.