Texas doesn't require routine testing of existing private wells. Keeping a well safe is up to the owner, so test for bacteria and nitrate every year.
Where can I get my well water tested in Texas?
Texas A&M AgriLife Extension's Texas Well Owner Network holds periodic county well water screenings, some free and some for a small per-sample fee. Otherwise, use a lab certified by Texas for drinking water. The state's list is linked on this page.
Where these numbers come from
How many people use wells: EPA estimates from the 2020 Census.
Lab tests: Water Quality Portal (USGS and EPA), well samples since 2000.
State programs and links: Texas Water Development Board and Texas Commission on Environmental Quality (TCEQ).