How to trim the fat on my log graphs

My log graphs are pretty insane sometimes because we converted our
repo from SVN and haven't had a chance to delete all of the remote
branches. We still have quite a few (maybe 20).

When I do `git log`, I am shown about 10-15 vertical lines and the
branch I currently have checked out isn't even at the top of the
graph, it's burried somewhere one or two pages down. I think this is
really confusing and makes my log virtually useless. Here is my log
command:

git log log --graph --abbrev-commit --decorate --date=relative
--format=format:'%C(bold blue)%h%C(reset)%x09%C(bold
green)(%ar)%C(reset)%C(bold yellow)%d%C(reset) %C(dim
white)%an%C(reset) - %C(white)%s%C(reset)' --branches --remotes

I realize the `git log` documentation has a whole section on trimming
history so it is more useful, but after reading that section I am left
completely confused and I have no idea which additional options I can
add to trim the fat on my logs.

Ideally what I want, for starters:
- The branch I'm on should be at the top of the graph (first commit in
the log graph should be the tip of my checked out branch).
- Commits in other branches that are not ancestors of the current
branch should not be shown.
- Merges *into* my branch from other branches should show the graph
line for that branch but allow me to specify a "depth", meaning for
example that ancestors that are 3 "parents" away
(great-grandchildren?) will not be rendered (the 3 is an example, I
could specify any number)

The goal is to weed out the really distant and irrelevant commits. I
really just want to see the commits that are closely related to my
current branch. Thanks in advance.