Why do females always run like this?
Looking at a WARNING! TvTropes WARNING! One thing is clear, this kind of running is an exageration. So you will not probably find examples in real world.

But what is it exageration of? That character is female and that she is a ditz. In proper "girly run" arms are close to body and move in controlled way. This (supposedly) allows control of bouncing breasts. I can clearly imagine woman IRL running like that if she doesn't want to have a bra accident. Plenty of women will tell you that breasts bouncing around is not comfortable.

But character that lacks motor control or physique might have problem with this kind of running style. Which is why they attempt it, but fail and their arms start flailing around. It might also be due to exhaustion, as fine motor control becomes more difficult.

Another point that just came to me is that, in real life, you mostly see people running who actually want to run. So they put effort into learning good running style. And in case of females, invest in good sports bra.