Nearly 8 years later but what the heck?
As I write, a crescent moon is setting in the west. The sun is below the horizon but the bright portion of the moon points towards it. Because the plane of the solar system (the ecliptic) is tilted, relative to the horizon, the lit part is from, roughly, the 1-o'clock to the 7-o'clock positions. The rest of the moon's face is lit by "earthshine" and is a pale grey, lighter than the sky's dark blue shade.
So the shape of a new moon is the *reverse* of "C", with the 'horns' of the crescent pointing at an angle to the left and slightly upward.
Winding back to the start of the day, the sun would have risen first, the moon some time behind it i.e always to the sun's left. Again, the lit portion of crescent respects the sun's position and the horns of the crescent would have angled left but downward. Overall, the brightness of the sun may have drowned out the light reflecting off the moon, for much of the day. If it fails to contrast adequately with the sky colour then you would struggle to perceive it against the glare from the sun.
The "C" shaped moon is the waning, old moon, which you'd only see if you were up before the dawn. Again, it is hard to spot against the sun's glare and it sets before the sun i.e. it is always to the sun's right.
So, for cartoon purposes, you can either: a
i) Use artistic licence, show a crescent rising, horns down, crossing the sky, all the while rotating clockwise, such that the horns are pointing left, at the zenith, then upwards as it sets or
ii) show the sunset, then have the cresent emerge from the increasing gloom but low above the horizon, then it sets but the rest of the passage of time would have to be portrayed by having the stars move and set, while preserving their patterns - which would be horribly complex to do by hand.
As above, it's a cartoon and nobody's going to be that fussed. In fact, the more kids discover that reality is unlike cartoons, the cleverer they feel and the more curiosity they have about why things are as they are.