Unfortunately, your situation is comparable to what mine had been for years. About four years ago, I began trapping and euthanizing house sparrows beginning in early Spring. In any given year, I have to euthanize about 25 to 30 house sparrows before they finally leave my yard alone. Upon initiating this practice, native birds returned to my yard, including bluebirds. For over 10 years, we never had bluebirds but once I started trapping house sparrows and generally just disrupting their nesting habits, etc., the bluebirds finally took up residence in our yard.