The recent cases aren't bribery in the classic sense, but Saudi princes (& in this case, very senior ones, from the innermost circle of power) using foreign purchases to divert money from their own budgets. It's Saudi money, transferred by people who are authorised, under Saudi law, to do so, but who are diverting it to uses they don't want to appear in their departmental accounts.
The famous BAe 'bribes' which the Saudi ambassador to the USA was involved in were Saudi money controlled by the ambassador. He's now head of the Saudi national security council, so it hasn't hurt him with his uncle the king, or his father, who is crown prince, deputy prime minister & minister of defence. That money was never paid to BAe as part of a contract, it was sent through BAe accounts from one Saudi account to another. It's very hard to characterise that as a bribe, however dirty it is.
You see the level this goes on at? We're not talking about bribing officials. The people at the very top of the Saudi government want to be able to shuffle around money, & use foreign firms to do so.
Much of the older stuff was corrupt in the classic sense, but indirect. The fine BAe paid, for example, wasn't for directly paying bribes, but making payments to intermediaries knowing there was a high probability of them being passed on to officials. That was the usual way.
State buyers insisted on doing business through their local agent, who charged very high fees. These fees were paid by the vendor, but added on to the contract price. After deducting a share (set by the buyer), the agent passed on most of his fee to the senior officials involved in the purchase.
The vendor (in theory) had clean hands, officially, knowing nothing of where the money went after they paid it, but in practice, everyone knew. Some very big names were involved as agents in this sort of stuff, owners of major London department stores, sponsors of buildings at Washington DC universities, etc.