Should teenage girls be allowed to wear bikinis on field trips involving swimming?

They should be free to either wear bikinis or not to wear them, as with anything else. I will also say this: if you have never swam in a sea, a lake or a river nude, then you have not lived.

In Nordic cultures it is not too uncommon for men and women, even when complete strangers, to be in the same sauna rooms totally nude. People there are not as ashamed of their bodies as in cultures more puritan in this regard, and no issues arise. Japanese also have a very relaxed relationship with public nudity, albeit there are some boundaries there you are not to cross, such as advertise your nudity on a big scale.