Heat the vegetable oil in a large skillet over medium-high heat.
Add the beef cubes in batches and brown them on all sides, about 5 minutes per batch. Transfer browned beef to the slow cooker.
In the same skillet, add the chopped onions and cook until softened, about 4 minutes.
Add the minced garlic and cook for 1 minute more, stirring frequently to prevent burning.
Stir in the sweet paprika, smoked paprika, ground caraway seeds, dried marjoram, black pepper, and salt. Cook for 1 minute to release the spices’ aroma.
Add the diced tomatoes with their juice, beef broth, water, and tomato paste to the skillet. Stir to combine and bring to a simmer.
Pour the tomato and spice mixture over the browned beef in the slow cooker.
Add the diced red and green bell peppers, sliced carrots, and bay leaf to the slow cooker. Stir gently to combine.
Cover and cook on low for 6 to 7 hours, or until the beef is very tender.
About 30 minutes before serving, mix the all-purpose flour with cold water to create a slurry.
Stir the slurry into the slow cooker to thicken the sauce. Replace the lid and cook for an additional 30 minutes.
Remove the bay leaf before serving.
Serve the beef goulash hot, garnished with chopped fresh parsley.