Preheat oven to 375°F (190°C).
Season the pork tenderloin with salt and pepper. Heat oil in a cast iron skillet over medium-high heat. Sear the pork on all sides until golden brown. Remove and let rest briefly.
In the same skillet, arrange the onions, turnips, and garlic. Place the pork tenderloin back on top of the vegetables. Pour half the ale over the mixture to baste.
Cover the skillet tightly with foil and place in the oven. Roast for 45–60 minutes, depending on thickness, until the pork reaches an internal temperature of 145°F (63°C).
Remove foil and broil for 3–5 minutes to crisp the top. Let the pork rest for 10 minutes before slicing.
While the pork rests, make the sauce. In a saucepan, combine lingonberry jam, the remaining ale, and honey. Simmer until thickened. If the sauce is too thin, whisk in a cornstarch slurry (1 teaspoon cornstarch mixed with 2 teaspoons water).
Slice the pork, serve over roasted vegetables, and drizzle generously with lingonberry ale sauce.