Fill a large saucepan with water, add a small handful of salt, and bring to a boil over high heat. When the water boils, add the linguine and cook according to the package directions for al dente pasta.
While the pasta is cooking, heat the oil in a large skillet over medium-low heat. Add the shallot and cook until transparent, 1 to 2 minutes. Reduce the heat to low, stir in the garlic and ginger, and cook until the mixture is aromatic and soft, about 4 minutes. Stir in the turmeric and cook 1 minute more.
Increase the heat to medium and add the shrimp. Cook, stirring constantly, until the shrimp are firm and pink all over, 4 to 6 minutes depending upon the size of the shrimp. Remove the skillet from the heat.
Drain the pasta in a colander, reserving a cup of the pasta cooking water. Return the pasta to the saucepan and stir in a splash of olive oil.
Return the skillet to medium-low heat. Stir in the yogurt and 1/2 cup of the reserved pasta water and cook until the liquid is fully incorporated, 3 to 4 minutes. Add more water to thin the sauce, if desired. Season to taste with salt and pepper.
Reduce the heat to very low. Add the pasta, arugula, and tomatoes and stir through until the arugula has wilted. The pasta will soak up the sauce. If it looks a little dry add a splash more pasta water until you have a sauce that coats every strand. Serve immediately.