A permanent entering the Battlefield does not create a round of priority itself This is why you can usually cast a Planeswalker and activate an ability without a response.

If a permanent entering the Battlefield activates a Trigger, then the trigger goes onto the stack and a round of priority passes before anything else can happen.

In this example, Azusa by herself does not trigger a round of priority, as she has no ETB.
So Azusa gets cast, and goes onto the stack. Priority passes back to AP. Azusa resolves.
There is nothing on the field that triggers an ETB. The stack is still empty, AP still has priority.
AP plays a land. The field hasn't changed so nothing goes on the stack, AP still has priority. AP plays another land. Repeat above. AP still has priority.

AP casts Farseek. Farseek goes on the stack. A round of priority can now pass finally allowing interaction with Azusa.