I uploaded two images. The top image show the first box power comes into,
or the service disconnect. The bottom picture is the modern service panel
which branches off from the service disconnect.

Service Disconnect:
Does it look like this busbar will be able to support 4 outgoing conductors?
1. #4 ground going to grounding rod(s)
2. #4 ground going to water pipe
3. Big gague mesh going to service panel neutral bus bar
4. #4 ground going to service panel ground bus bar

If I can't put more than one conductor on each bolt, then I'm guessing I'll
have to upgrade. Will I need to change the whole box or can I just upgrade
the bar?

Modern Service Panel:
Some connections have actually be corrected since this picture was taken,
but please comment as-is and I'll know what to change. You'll notice that
it's upside down. This is how it was installed. The #4 ground coming from
the service disconnect enters the panel and attaches to the LEFT bus bar.
It attaches to the bus bar in the same as you would install a wire for a new
circuit. Is it correct to assume this wire should have been installed to
the heavier bolt to the lower left of the bar?

Please identify at which point the ground bus grounds to the box.

http://www.geocities.com/mydeadpresidents/index.html

I've got a new electrician coming over in a day or two. Hopefully, this
will be my last post.... thanks for your patience.