How do betting sites update odds during a sporting match in real-time?

The collection of odds (which I will call 'the book') are set up so bookies will make a profit.

As the bets come in, those odds have to shift in response, to keep the book in profit, so a given bet placed at one time will get different odds than a bet placed at a different time.

A second factor: if bookies offer odds too different from other bookies, they may experience punters exploiting that by 'making a book' of their own. So they tend to pay very close attention to other bookmakers odds and adjust in response.

If a bookie gets a bet large enough to eat their profits if it wins, they may attempt to 'lay off' a chunk of the bet with other bookmakers.

In some sporting events, the use of point-spreads comes into the calculations, either in place of or along with over-round, depending on the situation.

These days the whole book can be managed by computers. (If they're working with point-spreads, there may be some more-or-less formal model for the point-distribution of the two sides that will update over time. It may include whatever factors it occurs to them to include. Such information tends to be secret, since telling people too much information about your models will make you more easy to exploit, and there's a lot of money involved.)

You may find the following details of some help Making a book and Spread betting.