iLrrr-n,
If you are just interested in producing american R, here is a tip from "American spoken english in Real life: fast natural, urgent survival foreign accent begone!" by DG Davies.
" r + vowel = ur + vowel. First say u. Tongue back into mouth does not move while lips smile changing to next vowel.
write - right urait, three thuri, brow buraun, etc."
Here is a video that follows the above advice:
Note the rounded ness of /r/