It probably has to do with the thermal diffusivity $\alpha = \kappa/(\rho c_p)$. It is equal to $\frac{\frac{\partial T}{\partial t}}{\nabla ^2 T}$ when convection and radiation effects are neglected. This latter formula gives some insights. The faster the temperature changes compared to its curvature (or the divergence of the gradient of T), the greater is the thermal diffusivity. So, from your description, all seems to point that the thermal diffusivity of the metal (steel?) is low enough to ensure that the cold side will still have an increase in temperature even about $15\ \mathrm{s}$ after the hot source was removed.
Also, I do not agree with the claim that temperature does not have inertia. For example, the heat from the Sun will diffuse through the Earth ground in such a way that it is possible to dig a few meters below the ground surface and be able to spot summer times and winter times. But the amplitude of the temperature variations decays exponentially with depth. At large depths (something like above 20 m or so) only the average surface temperature is still distinguisible. By digging deep enough, it is possible to know the average surface temperature up to hundreds of years or even more (I remember several articles about that, I might try to provide references later).
The above example is relevant, because it shows that even if the Sun was suddenly removed, the heat from past summers would still diffuse deeper and deeper into the ground, heating colder parts, well after the heat source (Sun) is removed. The same thing is probably happening from what you describe with the heated bar under a flame suddenly removed.
To summarize, you've noticed a peculiarity of conductive heat transfer. This is normal and would happen with any solid. The fact that the temperature increase lasted for about 15 s after the heat source was removed is due to both the geometry of the material and on its thermal properties, in particular its thermal diffusivity. A material with the same geometry than the steel rod but with a higher diffusivity such as copper would display a shorter such time, and conversely: a material with a lower diffusivity such as glass would display this temperature rise at their colder side for longer.