Target Practice and Marksmanship Training Support Act This bill amends the Pittman-Robertson Wildlife Restoration Act to facilitate the construction and expansion of public target ranges by: (1) authorizing a state to pay up to 90% of the costs of acquiring land for, expanding, or constructing a public target range; (2) authorizing a state to elect to allocate 10% of a specified amount apportioned to it from the federal aid to wildlife restoration fund for those costs; (3) limiting the federal share of those costs under such Act to 90%; and (4) requiring amounts provided for those costs under such Act to remain available for expenditure and obligation for five fiscal years. The United States shall be shielded from any civil action or claim for money damages for injury to or loss of property, personal injury, or death caused by an activity occurring at a public target range that is funded by the federal government pursuant to such Act or located on federal land, except to the extent provided under the Federal Tort Claims Act with respect to the exercise or performance of a discretionary function. The bill urges the Forest Service and the Bureau of Land Management to cooperate with state and local authorities and other entities to carry out waste removal and other activities on any federal land used as a public target range to encourage its continued use for target practice or marksmanship training.