How does the architecture prevent someone taking over by running a several billion clients? I'm new to bitcoin and was wondering: As far as I know, bitcoin relies that the majority of the users is to be trusted.

Why is it not a problem then if someone starts up billions of clients, eg. a data center owner? Seemingly a single entity could take over the whole network. What is the bitcoin feature that is the counter-measure here?
The majority needs to be trusted, but not weighted by their number of software instances, but by either computational power or economic power, depending on the context.

The main technical assumption is that the majority of hashrate is not trying to attack the network. So an attacker would need to purchase and operate more hashrate than the rest of the network combined; this isn't easy.