In his conversation with Lionel, Lahlou describes his daughters' success in school and sports in an interesting way. He uses the verb assurer:

Les deux grandes franchement, elles assurent.

The two older girls, frankly, they're doing great.

Elles assurent super bien à l'école, au sport.

They're doing really well in school, in sports.

Captions 82-83, Lionel & Lahlou - Être musulman

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Elles assurent is a familiar way of saying "they're doing great," "they're doing a great job." Lahlou also could have used the preposition en to specify what the girls are succeeding in: elles assurent en maths (they're good at math), elles assurent en natation (they're good at swimming). 

More often, assurer means "to assure" or "ensure," or, when reflexive, "to make sure"/"to check":

Je vous assure qu'elle est là.

I assure you that it's there.

Caption 37, Le saviez-vous? - Les verbes du 1er groupe les plus utilisés

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Mais on doit s'assurer que le pneu est bien assis

But we have to make sure/check that the tire is well-seated

sur la jante et ne pas trop gonfler.

on the rim and that we don't inflate too much.

Caption 19, Sports Shop - La mécanique d'un vélo

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BANNER PLACEHOLDER

Assurer has a few other meanings as well. It can mean "to secure" or "achieve":

Ses affiches et ses tableaux

His posters and his paintings

ont permis au Moulin Rouge

allowed the Moulin Rouge

d'assurer une notoriété rapide et internationale.

to achieve rapid international notoriety.

Captions 19-21, Amal et Caroline - Moulin Rouge

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Il est difficile d'assurer un emploi en ce moment.

It's hard to secure a job these days.

It can mean "to take care of," "handle," or "deal with":

Je dois assurer l'école.

I have to take care of the school.

Caption 13, Les zooriginaux - 3 Qui suis-je?

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La brigade des pompiers assure l'extinction des incendies.

The fire department takes care of putting out fires. 

Or it can mean "to insure," as in "to provide insurance coverage":

Notre maison est assurée.

Our house is insured. 

Likewise, its noun form assurance can either mean "insurance" or "assurance," or more precisely, "self-assurance," "confidence," "certainty":

Le stress au travail a en effet un coût, humain bien sûr,

Stress at work does indeed have a cost, a human one of course,

et économique pour l'assurance maladie:

and an economic one for national health insurance:

près de cinquante milliards d'euros.

close to fifty billion euros.

Captions 57-58, Le Journal - Le stress au travail

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Il faut parler avec assurance pour convaincre les gens.

You have to speak with self-assurance/confidently in order to convince people.

So whether you're assuring, ensuring, making sure, or insuring, assurer is the verb to use. You can find even more ways of using it here.