Start dragging before the first link

If you try to start your selection inside a link, the browser window thinks you're trying to drag that link to another document or folder. If you instead start before the first link, it'll let you select text and images, including links.

The mouse cursor shape tells you whether you'll be selecting or dragging links: if it's an arrow or text I-beam, you can select; it it's a hand, you're in drag land.

Start dragging above the first link, not to the left of it

Because of the way many web pages are formatted (using tables), you'll often find it easier to select links by starting just above the left edge of first one, not to the left of it.