Season the shrimp with kosher salt and pepper.
Heat 2 tablespoons of the oil in a large skillet over medium-high heat. When shimmering, add the shrimp to the skillet in a single layer and sear until cooked through and golden brown, about 2-3 minutes per side (you may need to do this in batches depending on the size of your skillet). Transfer cooked shrimp to a plate and set aside.
Add another tablespoon of oil to the skillet along with the broccoli florets and red bell pepper. Saute for 2 minutes, stirring, then add ¼ cup water. Continue sautéing the veggies until the water has completely evaporated and the veggies are tender, about 3-4 for minutes.
Add the remaining tablespoon of oil to the skillet and the spiralized sweet potato and cook, stirring, until just tender, about 10 minutes. Transfer the cooked sweet potato and veggies onto the plate with the shrimp and set aside.
If the skillet is dry, add 1 more teaspoon of oil along with the tablespoon of red curry paste. Saute the red curry paste until fragrant, about 1 minute. Pour in the coconut milk and whisk to combine with the curry paste. Bring to a simmer. Once simmering, add the fish sauce, then cooked veggies and shrimp back into the skillet and toss until evenly coated.
Stir in the fresh cilantro and serve immediately.