First dumb question: is it actually overheating (light comes on, possibly steam?) or is it just running hot?
Normal temp is fractionally below centre on the gauge, "9 o'clock" (we punters prefer it to be slightly below mid-point). The fan will try to hold it at 10 o'clock, and proper overheat is 11 o'clock. The fan doesn't really knock the temperature down, justs holds it at about 10 o'clock.
That it all sorts itself out as soon as you get moving again (and noting that the fan is cutting in) suggests two possibilities:
- With engine speed up, the pump works faster and is able to move the water around at an acceptable speed (I have a different car where the factory recommendation is to counter-intuitively rev the engine a bit to get the water moving. This doesn't work noticeably on the Punto, but explains the idea). The lack of sludge leads me to think that this isn't the problem.
- Fan or not, not enough air is passing through the radiator. The air side is blocked. Only when you've got a bit of speed up do you generate enough airflow to cool the temps.
There are a couple of checks on the latter. First, take a look through the air inlet in the bumper, just in case there's a plastic bag of obvious debris blocking it.
The second is the rare occurence of fin rot - the fins have broken away, not only costing cooling efficiency but also blocking airflow. When cold, run your hand down the back of the radiator. You would find broken fin debris there. It feels like rock salt, and if you break some away it is immediately obvious as aluminium shavings. If it is that, you just need to replace the radiator (try Aaron Radiators for a new one, or ebay or a scrappy). While the system's open, make sure the water side gets flushed through as well, just in case.
As you can probably guess, I had this rare problem (on a Punto GT). Took ages to identify, and was missed by several services. Symptoms were that it ran fractionally hot (just past 9) almost all the time, rapidly heated up (to 10, with the fan cutting in) if I hit urban or a motorway queue (and all got a bit nervy), and would cool back down pretty quickly as soon as I got back on clear roads. If this sounds like your symptoms, you should be looking at airflow. A new radiator would cure it, and the rest of the system was fine (as it should be, given that most of it had been replaced already!)