there is one route that I do know you may take. Write him/her a formal letter and mail to them certified with return receipt stating your concerns. Basically state that their birds are not monitored or controlled and continue to enter and trespass onto your property and causing excess damages to your lawn and other personal property and destroying your gardens, and also state that on such and such date you personally tried to talk with them with your concerns only to their obvious intent of avoiding any conversation with you by not answering door as you clearly saw them. And state that you have documented proof of their birds entering and damaging your property and continuing terrorizing your children to a point that their refusal and your unwillingness to allow your children outside due to possible dangers that turkey pose toward a human, namely that turkeys like to peck at shiny objects, also have been known many occasions to peck at human eyes when given the chance to do so. That this is your written attempt to resolve this matter out of court, Upon receiving this letter any failure to keep their birds off your property will result in legal action.
Don't bother stating exactly what your legal action would be, but to tell you, document everything, take video or pictures when they enter your property, take pictures of their birds when their birds are on their own property so they cannot deny the birds belong to them, take pictures of your damages. take them to small claims for the damages to your lawn resulting in grass seed planting,etc. sue also for the damages to your gardening, sue for the time to repair these damages and cleaning up the bird feces. and add a couple of thousand for the [personal suffering that this caused for your child. I believe this would repair your problem.