Rinse and core the cabbage. Carefully remove 10 outer leaves, place them in a saucepan, and cover with water. Bring to a boil, and simmer for 5 minutes. Remove and drain cooked cabbage leaves on paper towels.
Shred ½ cup of the raw cabbage, and set aside.
Brown ground turkey, ground beef, and minced onion in a skillet. Drain off the fat.
Combine meat mixture, bread crumbs, ¼ cup water, and black pepper in a mixing bowl. Drain the tomatoes, reserving liquid, and add ½ cup of the liquid to the meat mixture (set aside the tomatoes). Mix well.
To assemble cabbage rolls: Place ¼ cup filling near the edge of each cooked cabbage leaf. Roll up the leaf, tucking in the sides to keep the filling inside. Place folded side down in a large skillet; arrange cabbage rolls side-by-side.
On top of the assembled cabbage rolls, add tomatoes, sliced onion, 1 cup water, shredded cabbage, and carrot. Cover and simmer about 1 hour (or until cabbage is tender), basting occasionally.
Remove the cabbage rolls to a serving platter, and keep warm.
Mix lemon juice, brown sugar, and cornstarch together in a small bowl. Add to the vegetables and liquid in the skillet and cook, stirring occasionally, until thickened and clear. Serve over cabbage rolls.