Vorschub leisten
to encourage
To support, promote, or favor an action or development, usually a negative one.
Man darf der Korruption keinen Vorschub leisten.
One must not encourage corruption.

support / feed
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to encourage
To support, promote, or favor an action or development, usually a negative one.
Man darf der Korruption keinen Vorschub leisten.
One must not encourage corruption.

The word is made up of 'vor' (forward) and 'der Schub' (the push/thrust). The literal meaning of a 'push forward' helps you understand both meanings: the figurative one and the technical one.
Der automatische Vorschub der Säge muss exakt eingestellt sein.
The automatic feed of the saw must be set precisely.
The figurative meaning exists almost only in the fixed expression '(einer Sache) Vorschub leisten' (to encourage/promote something). The noun 'Vorschub' is rarely used on its own here. The thing being promoted is in the dative.
Mit seiner unklaren Kommunikation leistete er den Gerüchten Vorschub.
With his unclear communication, he encouraged the rumors.
The expression 'Vorschub leisten' (to encourage/promote) is used exclusively for negative developments or actions. Typical examples are 'der Korruption Vorschub leisten' (to encourage corruption), 'dem Extremismus Vorschub leisten' (to encourage extremism), or 'dem Missbrauch Vorschub leisten' (to facilitate abuse).
Man wirft der Regierung vor, der Spekulation Vorschub zu leisten.
The government is accused of encouraging speculation.
The meaning related to tool movement is a technical term from technology, especially manufacturing and mechanical engineering. Outside this context, 'Vorschub' (feed) is not used in this sense.
Eine hohe Vorschubgeschwindigkeit kann das Werkzeug beschädigen.
A high feed rate can damage the tool.
support
'die Unterstützung' and 'der Vorschub' can both mean that something is promoted or encouraged. 'Unterstützung' is neutral and usually positive: you help a person, a project, or a cause. 'Vorschub' is almost always used in this meaning in the fixed phrase 'etwas Vorschub leisten' (to encourage something / give something a boost). It means that you foster a development, often something negative such as rumors, fear, or conflict. Learners may confuse the words because both have to do with promoting or supporting something. Remember: people receive 'Unterstützung', while you 'leistet Vorschub' to negative developments.

Mit solchen Gerüchten leistet man dem Misstrauen nur Vorschub.
With rumors like that, you only encourage mistrust.
Für den Sprachkurs suchen wir noch Unterstützung von ehrenamtlichen Helfern.
We are still looking for support from volunteer helpers for the language course.