
mäßigen
to moderate
To reduce or dampen the intensity, extent, or speed of something.
Der Arzt riet ihm, seinen Alkoholkonsum zu mäßigen.
The doctor advised him to moderate his alcohol consumption.
to moderate / restrain
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to moderate
To reduce or dampen the intensity, extent, or speed of something.
Der Arzt riet ihm, seinen Alkoholkonsum zu mäßigen.
The doctor advised him to moderate his alcohol consumption.

to restrain oneself
To hold oneself back and control one's behavior or remarks so as not to seem excessive or inappropriate.
Bitte mäßige dich in deiner Wortwahl, wir sind hier nicht unter uns.
Please restrain yourself in your choice of words; we're not among ourselves here.
The verb 'mäßigen' (to moderate/restrain) belongs to the word family of 'das Maß' (measure) and the adjective 'mäßig' (moderate). Literally, it means bringing something to a healthy or appropriate 'Maß' (measure).
Versuche, deinen Kaffeekonsum ein wenig zu mäßigen.
Try to moderate your coffee consumption a little.
The meaning depends on how it is used. Without a reflexive pronoun, it means reducing an external thing ('das Tempo mäßigen' — to moderate the pace). With a reflexive pronoun ('sich mäßigen' — to restrain oneself), it means controlling one's own behavior.
Der Redner wurde gebeten, seine Lautstärke zu mäßigen.
The speaker was asked to moderate his volume.
You can 'mäßigen' (moderate) abstract things or behaviors. Typical combinations are 'das Tempo mäßigen' (to moderate the pace), 'den Alkoholkonsum mäßigen' (to moderate alcohol consumption), or 'seine Begeisterung mäßigen' (to moderate one's enthusiasm).
Bitte mäßigen Sie Ihre Geschwindigkeit in der Spielstraße.
Please moderate your speed in the traffic-calmed street.
When a person is supposed to hold themselves back, the verb is used reflexively: 'sich mäßigen' (to restrain oneself). The reflexive pronoun is in the accusative.
Der Trainer sagte dem Spieler, er solle sich in seiner Kritik am Schiedsrichter mäßigen.
The coach told the player that he should restrain himself in his criticism of the referee.
to reduce
Both words refer to reducing intensity or extent. „Reduzieren“ is neutral and can describe any planned change in amount, whereas „mäßigen“ usually dampens a level that is perceived as too strong or excessive. „Reduzieren“ is appropriate for an objective statement about an amount; when something is too intense, „mäßigen“ is more suitable.

Wegen der Beschwerden mäßigen wir die Lautstärke der Musik.
Because of the complaints, we moderate the volume of the music.
Wir reduzieren die Lautstärke der Musik auf die Hälfte.
We reduce the volume of the music by half.