to push through / to assert oneself / to permeate
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to push through
To realize or bring something (a plan, demand, idea) into general acceptance despite resistance.
Die Regierung konnte das neue Gesetz im Parlament durchsetzen.
The government was able to push the new law through parliament.
This verb is separable or inseparable depending on the meaning. In the meaning 'einen Plan verwirklichen' (to realize a plan) or 'sich behaupten' (to assert oneself), it is separable, with stress on 'durch'. In the meaning 'etwas durchdringen' (to penetrate something), it is inseparable, with stress on 'setzen'.
Sie setzte ihre Idee am Ende durch. (trennbar)
She pushed her idea through in the end. (separable)
In the meaning 'sich behaupten' (to assert oneself), the verb is always used reflexively: 'sich durchsetzen'. The opponent or difficulty is often named with the preposition 'gegen' (against).
In der Diskussion konnte er sich nicht durchsetzen.
In the discussion, he couldn't assert himself.
When 'durchsetzen' (to push through) is used transitively and separably, it usually has an abstract object. Typical nouns are: einen Plan (a plan), eine Forderung (a demand), Interessen (interests), eine Reform (a reform), or den eigenen Willen (one's own will).
Die Gewerkschaft will höhere Löhne durchsetzen.
The union wants to push through higher wages.
In the inseparable meaning 'durchdringen' (to penetrate), the verb is very often used in the passive with the preposition 'mit' (with). The structure is: 'etwas ist durchsetzt mit etwas' (something is permeated with something).
Seine Rede war mit vielen Zitaten durchsetzt.
His speech was permeated with many quotations.
to carry out
'durchführen' and 'durchsetzen' can both mean that a plan eventually becomes reality. However, 'durchführen' mainly describes organizing and doing a planned action. 'durchsetzen' emphasizes that there is resistance and that someone still successfully prevails. That is why you can 'eine Veranstaltung durchführen' (carry out / hold an event) without 'etwas durchzusetzen' (pushing something through). You are more likely to 'eine Forderung, Regel oder Idee durchsetzen' (push through a demand, rule, or idea). The risk of confusion arises because both verbs have to do with success and implementation. Pay attention: 'durchführen' means to carry out, while 'durchsetzen' means to achieve something despite resistance.
Die Mitarbeiter setzen ihre Forderung nach flexibleren Arbeitszeiten durch.
The employees push through their demand for more flexible working hours.
Im Büro führen wir heute eine kurze Schulung für das neue Programm durch.
In the office, we are holding a short training session for the new program today.