The urbanmech is about the exact speed of an abrams off road, and it takes an abrams about 18 seconds to spin around safely. The urbanmech doing the same is about right. A lot of the turns your describing sound like torso twists, which mech can do seemingly in almost no time at all. There is a way to turn faster in a mech... falling and standing. That can spin you around real quick, but for 'safe' turns once standing the mp cost in 10 seconds is good.
Also note, the video games depict mechs not under throttle as being immobile. This is not the case. Mechs, even when spending 0 mp, even when prone, are still moving within the 30 meter hex constantly. So an 'immobile' standing still turn is not what the game represents when you spend MP to turn.
Also, turning 60 degrees at speed in a short period of time ISNT easy. Indoor tracks with steeper turns versus outdoor tracks, or flat 400m tracks versus slightly banked tracks make a difference in speed, and a 60 degree turn in 30 meters is sharper of a turn then the tracks humans with little legs run on, so the effect would be much more noticeable for mechs. There is good reason mechs can slip turning 'just' 60 degrees moving on pavement, that is pretty solid as I understand. The skid giving extra movement is all wrong and incorrect, but that comes after the skid fall.
Finally, the scale messes up preceptions. Sure, you, a human, can spin in a circle pretty fast. But what is your angular velocity? Its not very much, as our radius is small. But a mech has a much much wider radius just porpotionaly to a human, then multiplied by the 6 times taller bit. So a wide person at .2 meters radius spinning 1 time in 1 second is looking at 1.23 m/s speed. A mech, at human proportion which isnt even close to accurate with the wide bodies/shoulders mechs have, would have a 1.2 meter radius and 7.5 meter speed. So a mech, turning AS FAST as a human doing a 360 in one second, would be turning at 60 degrees per second. A more realistic wide mech would be half or a third of that, as mechs are far far wider then humans. Scale messes with perceptions, and a simple turn for a human is 6x slower for a mech.
This is why a giant godzilla walking slowly seems cumbersome as viewed from a distance, but then you calculate the speed its actually moving and it turns out its actually very very fast.