I am looking for ways to get around the Square-Cube Law for the giants, dragons, and other large creatures of my fantasy setting. What are some realistic ways that it could be done?
On top of what Keith Morrison suggested in their answer, you can also use a different medium where to have the organism.
Look at the big cetaceans: their size would make it impossible for them to be land dwellers, however living in water and taking benefit from the increased buoyancy it gives with respect to air makes the trick of not overloading their structures.
It is a simple fact of physics: strength is proportional to cross section, mass proportional to volume (this is more important than the relation of surface area to volume: heat management has a lot more flexibility).
In order "get around" it by physical means (ie, not magic) you have to do one or both of the following things: different materials allowing greater strength than mundane analogues, or significantly reduced mass and thus overall density. That's pretty much it. You can futz around with some designs so your giants don't actually look like merely enlarged humans any more, as an example, but in the end it comes down to the material aspect. You want to do somethign bigger, you need to change the meat and bones.
Lower gravity = less gravitational effect on things = things can be more massive.
High atmospheric / fluid pressure
If you have a dense enough atmosphere then you can have land whales. High pressure gives support to a creature's body.
Let animals use gravity emitting organs to cheat
More esoteric. What if very large animals had naturally evolved gravity/anti-gravity organs or graviton emitting / graviton reflecting cells?
What if dragons developed gravity-reflective scales covering their wings?
So what if all life were nanite-assisted mechanical hybrids but no one knew? Like, what if alien nanites got into the evolutionary chain somehow billions of years ago and then it just evolved like that? The machines could be capable of constructing bones made from carbon nanotubes and similarly upgraded muscles / neurons / etc. All without anyone being any the wiser.