Why must electrostatic field be normal to the surface at every point of a charged conductor ?

-- If the field is not perpendicular to the surface of the charged
conductor, then there's some horizontal component of the field there. 

-- If there's any horizontal component of the field along the surface,
then the charges will move in response to the field, until the horizontal
component is zero.  They're free to move since we're talking about the
surface of a conductor.

-- So if there's ever a horizontal component of the field along the
surface, things quickly adjust themselves to make it zero, and the
field ends up perpendicular to the surface again.

You would need to take a picture so i can see