Press your lips together. Air flows through your nose. Vocal cords vibrate.

Americans pronounce mouthguard as MOWTH-gard (/ˈmaʊθˌgɑrd/). Stress falls on the first syllable — keep everything else short and quick. You'll hear it in sentences like "The mouthguard protects his teeth during contact sports".
Record yourself saying "mouthguard" and play it back. The mic stays on your device — nothing's uploaded.
2 syllables, 6 sounds. Tap a syllable to jump to its row, then explore each sound's mouth shape and how it's made.
Press your lips together. Air flows through your nose. Vocal cords vibrate.

Start with a dropped jaw and flat tongue. Glide into a relaxed, slightly rounded lip position as the back of the tongue stretches up.
Place the very tip of your tongue slightly between your teeth. Blow air gently around it without voicing.

Raise the back of your tongue to touch the soft palate. Add vocal cord vibration, then release.

Open wide for the 'ah' vowel. Lift the tongue back and up while flaring the lips for the 'r'.
Touch the tip of your tongue to the roof of your mouth just behind your teeth. Add vocal cord vibration as you release.

The textbook way isn't wrong — it's just not how anyone actually says it.
In "mouthguard", the "d" is not released — the articulators get into position but hold without the burst of air. Air stops but there's no release burst — the articulators hold position.
Stress falls on the first syllable, not the others. Stretch MOWTH — keep everything else short and quick.
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.