Can Aesop's Pawnshop Remove Wyldside before the start-of-turn resolves?
You choose the order they resolve.

If Wyldside resolves first:


Draw two cards, lose a click
Sell Wyldside with Aesop's: trash it and gain three credits


Or, if Aesop's resolve first:


Sell Wyldside for three credits
Wyldside does nothing because it's been trashed (so you keep all four clicks)


This specific scenario is actually covered in the rulebook (PDF) on page 22, under "Simultaneous Effect Example":


  The Runner has Aesop’s Pawnshop and Wyldside installed and both have the same trigger condition of “When your turn begins.” The Runner begins his turn and can choose to trigger the optional conditional ability on Aesop’s Pawnshop first, gaining 3 [Credits] by trashing Wyldside. This stops Wyldside’s required conditional ability from triggering, keeping the Runner from losing [Click].