We report the results of an investigation of ambipolar transport in a quantum well of 15 nm width in an undoped GaAs/AlGaAs structure, which was populated either by electrons or holes using positive or negative gate voltage $Vtg$, respectively. More attention was focussed on the low concentration of electrons $n$ and holes $p$ near the metal–insulator transition (MIT). It is shown that the electron mobility µ increases almost linearly with increase of $n$ and is independent of temperature $T$ in the interval 0.3 K–1.4K, while the hole mobility µ depends non-monotonically on $p$ and $T$. This difference is explained on the basis of the different effective masses of electrons and holes in GaAs. Intriguingly, we observe that at low $p$ the source–drain current ($ISD$)–voltage ($V$) characteristics, which become non-linear beyond a certain $ISD$, exhibit a re-entrant linear regime at even higher $ISD$. We find, remarkably, that the departure and reappearance of linear behaviour are not due to non-linear response of the system, but due to an intrinsic mechanism by which there is a reduction in the net number of mobile carriers. This effect is interpreted as evidence of inhomogeneity of the conductive 2D layer in the vicinity of MIT and trapping of holes in ‘dead ends’ of insulating islands. Our results provide insights into transport mechanisms as well as the spatial structure of the 2D conducting medium near the 2D MIT.