Yes, both words are used in American English. And yes, we can distinguish both of them by pronunciation. The difference in pronunciation is similar to that of "rut" and "rot"; however, if you have trouble telling them apart by pronunciation, you'll have to rely on context - they're not very similar in meaning so they shouldn't be hard to tell apart.

crutch: the phonetic alphabet representation is /krΛʧ/ (Λ is the same as the “u” in “but”) crotch: the phonetic alphabet representation is /kraʧ/ (a is the same sound as the “o” in “pot”) Do you know the phonetic alphabet? It teaches the sounds of English.

Just as an interesting aside, in Australia we crutch sheep, which is the removal of wool from around the tail and between the rear legs of a sheep. This prevents fly-strike in an area likely to be contaminated with urine and faeces. If this isn't done the flies lay their eggs in the soiled area and the larvae can burrow into the sheeps flesh.

Hi Johnny Lin: Your question should say: Does the word "crotch" exist in American English, or is it just used in British English?" By the way, a "crutch" is a device that helps people walk when they have injured a leg; "crotch" means the junction or joining point of two limbs (person, tree, etc.).