Exercises for pronouncing the r

Solution 1:

The American English /r/ phoneme is rounded, so there should be some "W" sound.

The problem is likely insufficient retroflexion. Make sure your tonguetip is curled up and back, so that you can touch your hard palate with the bottom of the tonguetip. With your tongue in this position and your lips rounded, say [ə]. In practice, native English speakers don't actually touch anything with their tongue when saying /r/, but the tongue is almost there and that's the position, so you can practice it.

For further phonetics, let me recommend J. C. Catford's helpful little book A Practical Introduction to Phonetics; it's designed for the autodidact and is full of little phonetic experiments you can do.

Solution 2:

You have a case of rhotacism and two methods to correct the issue is described here or google Exercising the Rhotacism in Absence of Pathology