The US work place is not kind to introverts. In the US the aim is to stand out. Those who stand out get the promotion they feel they deserve, so speak up at meetings, offer ideas, do things with out being asked, be enthusiastic, be positive, do more than your job requirers. This won't work in the UK. Standing out in the UK involves blending in, keeping your head down and being a safe pair of hands.
Sucking up to management will get you exploited rather than promoted. Building the right reputation is about being ambitious but not too ambitious, enthusiastic but not over enthusiastic, confident but not over confident, positive but not idealistic and realistic not naive. In other words it's complicated.
Sometimes it's easier to get recognition outside your organisation and in so doing raise your profile inside, for example through representing your organisation in a multiagency forum or writing for the professional press.
Getting on a corporate project group is a good way of raising your profile in the organisation.
It's not a question of competence but self promotion.
However it better to have other people saying what a good job you are doing than you saying it about your self.
There is a view that introverts tend to be overlooked when it comes to promotion. In my experience its pessimists and cynics senior managers avoid.