Lift your bottom lip to touch the very bottom of your top front teeth. Blow air through this contact point without voicing.

Americans pronounce fourth as FORTH (/fɔrθ/). You'll hear it in sentences like "The fourth month is the birth of the earth".
Record yourself saying "fourth" and play it back. The mic stays on your device — nothing's uploaded.
1 syllable, 3 sounds. Explore each sound's mouth shape and how it's made.
Lift your bottom lip to touch the very bottom of your top front teeth. Blow air through this contact point without voicing.

Start with the 'aw' jaw drop and rounded lips. Pull the tongue back and up while keeping the lips rounded for the R.
Place the very tip of your tongue slightly between your teeth. Blow air gently around it without voicing.

The textbook way isn't wrong — it's just not how anyone actually says it.
Americans use a relaxed retroflex R — the tongue curls back rather than rolling. The R is one continuous sound with the vowel before it, not two separate sounds.