There are several clues that people in Braavos all know who the people of the many faced god are, and fear them (or at least, want to be thought well of by them):
When Arya was in Braavos last season, and was threatened by some street bravos, they slink away on recognising one of the faces of the kindly man who takes Arya in. This suggests that not only are they feared, but that the faces they use when in the public areas of the temple (the kindly man's old man face, the waif's serious girl face) are well known locally.
The fact that just showing anyone from Braavos a coin indicating that you're affiliated with them leads to that person doing whatever you ask to help you.
There's a little more on this in the books:
After Arya shows the coin, everyone from Braavos on her ship makes a point of being over-friendly and reminding her of their names. It's not confirmed why, but we later see that Faceless Men refuse assassination contracts if they "know this man"
- A small one from the books:
A few Braavos civilians are enlisted to take on various faces/personas of Arya in menial roles as part of her training (for example, a trader is enlisted to teach her to sell cockles and then treat her like any other cockle seller until the temple wants her for something else). These people seem to understand that this is part of some mysterious but important temple business and obey without question or comment, no matter how strange or mundane the task.
- The many-faced-god's temple is known as the "House of Black and White", and in these scenes the girl stands out as the only person on the streets wearing black and white (well, if you look closely it looks like very very very dark blue, but it was hard to find black dye in those days...). Another reason to think people would know who she was, or a least, who she represented.
So it makes sense that the most common faces the temple people use are well known, and that people in Braavos are well accustomed to people from this particular temple doing strange things, and know better than to interfere.
This explains not only why they don't intervene, but why they barely react at all - not even fearful watching or curiosity. "This is some sort of temple business. Don't look, don't react, don't get involved".
Everyone wants to be in the good books of the temple full of deadly face-changing assassins.