> OK, I have my personal zlib in $HOME/bin/zlib-1.2.5/
> , and yes there is a link libz.so -> libz.so.1.2.5
> in that directory. and some more of it:
> ./configure --with-zlib=$HOME/bin/zlib-1.2.5

If you do
--with-zlib=$HOME/bin/zlib-1.2.5

then you'll need these two directories:
$HOME/bin/zlib-1.2.5/lib
$HOME/bin/zlib-1.2.5/include

Do they exist ?