The US has
"precincts," which, according to Wikipedia, average around 1100 or so voters each -- urban precincts being, in general, larger and rural precincts being, in general, smaller. The equivalent of a "polling clerk" in the US would be an "election judge."
As for the waiting times, remember that US elections typically are consolidated, so that there are many races being voted upon at the same time. Unlike European elections, where a voter usually casts one or two votes at most, in the US it wouldn't be uncommon to vote in 20 or 30 different contests. See
this 2004 post for a more complete explanation.