Why XOOPS is heavy site? (Sooner is loaded)
is the xoops core reason?
Or theme?
What to do for a lightweight off-site?

There are so many different things that are causing it. Without knowing your Website is hard to guess.

You can try with setting up Cache for your modules.

You can also get a faster server.

But before you do it, you can also use some of the tools that will pinpoint you to your bottlenecks:

Most of the sites were made with XOOPS, Are heavy.

Xoops sites through the latest 2.5 versions have been tied down to older code base that has pieces going back to early PHP4 and don't take advantage of more recent php5. If you are talking about xoops sites in general being "heavy" or acting slow this is the reason.

With Alpha 2.6 the core is MUCH faster do to replacing much of the old core code and makes use of newer php5 features. When 2.6 eventually gets done it will make a huge difference compared to older versions.

Themes and modules used can also slow down a system.