Most fire departments have something like this:

Okay, but look how short that is compared to the American equivalent:

More specialized departments close to industrial facilities, airports can be also much bigger. This one is currently the biggest weighting 52 tons.

Okay, but look how short that is compared to the American equivalent:

The longer the truck is, the larger the turning radius it needs at intersections. The larger the intersections are, the faster regular cars drive through them. The faster the cars drive, the less safe it is for everybody else.

Deciding how large a vehicle a street should accommodate is called choosing the design vehicle. You pick that, and then the whole street is designed around it.

Guess what: here in the US, we often send even trucks like the second one I pictured -- the one that's even longer than your "industrial facility and airport truck" -- to residential neighborhoods. Fire departments want to own trucks like that and we just fuckin' let them. And that's why our neighborhood streets are too often designed like goddamned airport runways!

Edit: Oh, and by the way...

I agree that the US have way too many way too big trucks but this…

Ambulances and firetrucks in Europe and Asia are smaller than most american pickup trucks.

… is just wrong.

The MAN TGM 18.330 you cited has a wheelbase of 3,900/4,200/4,500 mm (source).

A Ford F-150 Super Crew with an 8' bed and an F-250/F-350 Crew Cab with an 8' bed, both of which are considered pretty typical American pickup trucks, have wheel bases of 163.7" (4158mm) and 176.0" (4470mm) respectively (source).

He's playing a little fast and loose with the notion of "most," but otherwise, no, he's actually not wrong!