Lag screws can't enter lag shields
Traditionally, those were made of lead. They didn't so much expand as squish outward against the concrete, engaging with it on a micro level. 

These newer pot metal ones can be a pain. They aren't really intended to expand much either, but because they're so hard it's difficult to drive the screw in. You just have to get rowdy with your wrench. It may also help to run the drill bit in and out a few times to make sure you have a fully round hole with no debris left inside.

If you find that you're breaking screws off you may want to try another brand or type of anchor.