Bring a large pot of water to a boil. Add the pasta and cook until al dente according to package instructions.
While the pasta is cooking, heat 1 tablespoon oil in a large pot over medium-high heat. Add the onion and cook for 3 minutes, stirring often until softened, about 3 minutes.
Add the garlic and cook, stirring, for 1 additional minute.
Stir in the tomatoes, broth, tomato paste, the oregano, red pepper flakes, salt, and black pepper. Simmer on medium low heat until the sauce thickens slightly, 8 to 10 minutes.
Add the shrimp and spinach and cook, stirring, for 5 minutes, until the shrimp is just cooked through.
Remove the shrimp mixture from the heat and add the drained cooked pasta to the sauce, tossing until well combined.