In a 12-inch nonstick skillet (avoid smaller skillets, otherwise ingredients may not fit), add 2 tbsp olive oil and diced turnips over medium-high heat. Cook until they're tender and golden-brown in some spots, 5 to 10 minutes, stirring occasionally.
Add corned beef, bell pepper, onion, garlic powder, salt, and black pepper. Stir together for 1-2 minutes until well-mixed.
For a crisper finish, gently press the mixture into an even layer, spread over the whole surface of the skillet. Cook until the bottom is crisp, 2 to 5 minutes, depending on how browned you prefer it to be. Turn off the heat.
Use a large spatula to scrape up a portion of the hash and overturn it onto a serving plate, repeating until it is all served.
Drizzle 1 tsp olive oil over the now-empty skillet. Crack eggs over the skillet, and cover with a lid. Cook on medium heat until the whites are cooked but the yolks are still liquid (or to your preference), a few minutes.
Top the hash with fried eggs. Prepare additional toppings as desired, such as salsa, sliced avocado, or shredded cheese. Serve.