We study the effect of disorder and temperature on the field-driven dynamics of a transverse domain wall occurring in thin and narrow magnetic strips made of a soft magnetic material such as permalloy. Motivated by a micromagnetic description of such a domain wall, we construct a model based on two coupled flexible lines enclosing the domain wall transition region, capturing both the finite width and the characteristic V-shape of the wall. Disorder is included via randomly distributed pinning centers interacting with the two lines. We study the field-driven dynamics of the domain wall in disordered strips in a finite temperature, and compare our findings to experimental observations of stochastic domain wall dynamics.