Tar a directory but don't store full absolute paths in the archive

Tar a directory but don't store full absolute paths in the archive

Part of a backup shell script, I have the following command :
tar -cjf site1.bz2 /var/www/site1/
When I list the contents of the archive, I get :
tar -tf site1.bz2
var/www/site1/style.css
var/www/site1/index.html
var/www/site1/page2.html
var/www/site1/page3.html
var/www/site1/images/img1.png
var/www/site1/images/img2.png
var/www/site1/subdir/index.html
But I would like to remove the part /var/www/site1 from directory and file
names within the archive, in order to simplify extraction and avoid
useless constant directory structure. Never know, in case I would extract
backuped websites in a place where web data weren't stored under /var/www
For the example above, I would like to have :
tar -tf site1.bz2
style.css
index.html
page2.html
page3.html
images/img1.png
images/img2.png
subdir/index.html
So that when I extract, files are extracted in the current directory and I
don't need to move extracted files afterwards, and so that subdirectory
structures is preserved.
There are already many questions about tar and backuping in stack overflow
and at other places on the web, but most of them ask for dropping the
entire subdirectory structure (flattening), or just add or remove the
initial / in the names (I don't know what it changes exactly when
extracting) but no more.
After having read some of the solutions found here and there as well as
the manual, I tried :
tar -cjf site1.bz2 -C . /var/www/site1/
tar -cjf site1.bz2 -C / /var/www/site1/
tar -cjf site1.bz2 -C /var/www/site1/ /var/www/site1/
tar -cjf site1.bz2 --strip-components=3 /var/www/site1/
But none of them worked the way I want. Some do nothing, some other don't
archive subdirectories anymore...
It's inside a backup shell script launched by a cron, so I don't know well
which user runs it, what is the path and the current directory, so always
writing absolute path is required for everything, and would prefer not
changing current directory to avoid breaking something further in the
script (because it doesn't only backup websites but also databases, then
send all that to FTP etc.)
How to achieve this ? Have I just misunderstood how the option -C works ?
Thank you very much for your answers and have a nice sunday.