I see the library site uses different techiques for different pages. The site uses "frames". The panel on the left side is a frame and the wider panel on the right site is a frame where most of the links listed in the left panel display. I noticed two exceptions, upon briefly looking at the site. The link for Find-It-Yourself opens a page with no frames, within the same browser window. And, the Your Account link opens a separate browser window. I can open Your Account by a normal left click, or right clicking and selecting Open or Open in New Window.
I believe that's the way it's supposed to work. I don't know why the link for Your Account will not open for you by just left clicking on it. I think I would ask someone at the library about this. They may have an answer as to why. And, I'd check the Basic Microsoft Internet Explorer troubleshooting page posted by one of the moderators, although I don't see anything there that looks like a fix, at least for what you're experiencing with the library site.
The case of the library site seems like a minor problem. I'm wondering what sites you've found with more imposing limitations, such as the where "Open" or "Open in new window" command are greyed-out.