It’s been my understanding that the numerous pirate crews in the Stepstones are supported or atleast tolerated by either Myr, Tyrosh or Lys. Is that correct, and if it is, why do the other powers in the regions like Braavos or Volantis tolerate that?

ethicalnecromancer:

racefortheironthrone:

Because Braavos and Volantis (and Pentos) also make use of pirates and/or sellsails (and the difference between them is about as clear as between pirate and privateer). 

I thought Braavos had its own standing navy. Isn’t that what the Arsenal shipyards are for?

It has both.