You will never see a good strongman with small forearms. The sport itself is grip intensive, and being able to distribute a massive amount of force through your hands is key to success.
If you watch strongman contests, you'll notice that every piece of equipment is supersized. It's the larger equipment (not our feverish masturbation habits) that give us the superhero-like forearms of Hellboy.
In order to build strong and powerful forearms, you have to do some serious grip training. One of the easiest ways to remember how to implement grip training in your program is to think "go big or go home." Using a larger piece of equipment will force your hands to activate.
Use a thick bar, thick dumbbells, grenade-style grips, Rolling Thunder handles, or thick attachments for all the exercises you do. Even Farmer's walks, an already grip-challenging exercise that's a staple in strongman, can be made more intense by increasing the diameter of the handle you're using.
I like to start out training with a larger apparatus in order to add as much grip work as possible, then switch over to a conventional barbell once my grip gives out.
I use an Axle bar for cleans, deadlifts, rows, and the occasional curl. Using a 2-inch axle will force you to squeeze harder and thus force those puny forearms to grow.
I use thick handles and grenades for chin-ups and sometimes unilateral pulldowns to isolate my weaker non-dominate limb.
Note that all of the above movements are flexion of the fingers and forearms. For a well-balanced forearm don't neglect extension of the fingers and forearms to counter all the flexion work you do. Two of my favorite moves for the extensors are rubber band extensions of the fingers and pushing my hand into a bucket of sand and then extending my fingers.