Thermally driven flows of superfluid 4He display unique features, often related to the presence of quantized vortices-line singularities embedded in the liquid. Here we focus on turbulent round jets, experimentally investigated using the flow visualization and second sound attenuation techniques, at Reynolds numbers exceeding 1e4. These turbulent flows are driven by releasing heat into a small volume of liquid, open to the surrounding bath through a cylindrical nozzle, 2 mm in diameter. Our measurements reveal in unprecedented detail how the tangle of quantized vortices associated to the jets arranges itself in space, for distances ranging from 9 to 34 nozzle diameters, at fluid temperatures between 1.64 and 2.10 K. We specifically find that the vortex tangle spreads in the radial direction, while it dilutes away from the nozzle. Additionally, the tangle density is found to systematically depend on the flow forcing. Two physical interpretations of the observed behavior are proposed, which could motivate further investigations of this peculiar flow. One leads us to conjecture a self-similar functional form of the vortex tangle density across counterflow jets. The other suggests that the position of the superfluid stagnation point-a characteristic feature of counterflow jets-could depend on the flow forcing as well.