Two-dimensional numerical models of the solar transition region are calculated using an inverse coordinates method which attains pressure equilibrium between the network magnetic field and the external comparatively field-free gas. If A(y, z) is the magnetic potential (a scalar in 2D), which is constant on field lines, the method involves interchanging dependent and independent variables to obtain a quasi-linear PDE for y(A, z), which is solved iteratively. The advantage of this approach is that magnetic field lines, including any magnetic interface, become coordinate lines, thereby simplifying the energy equation and free boundary problem. In order to examine the effects of self-consistent geometry on the thermal structure of the transition region network, we calculate four models. The energy balance includes the effects of radiation, conduction, and enthalpy flux. It is confirmed that the lower branch of the emission measure curve cannot be explained within the single fluxtube model if the classical Spitzer thermal conductivity is used. However, by including a turbulent thermal conductivity as proposed by Cally (1990a), transition region models are obtained for which the resulting emission measure curves exhibit the correct behaviour, including the observed turn-up below about 200 000 K. In summary, the broad conclusions of previous non-turbulent 2D models are confirmed, but most importantly, the turbulent conductivity hypothesis tested in 1D by Cally is shown to produce excellent agreement with observations in the more realistic geometry.