No, "Illegal bribery" is illegal. There is no such thing as "legal bribery". Some ignorant pop-culture or union educated Americans rely on emotion instead of rational thought to figure these things out. The Sierra Club is a worthwhile organization that saves the world but the NRA is evil or vice versey. You can't assume that lobbyists are criminals just because you have an emotional attachment to an issue. Make a complaint if you think you need to blow the whistle on a politician but the lobbyist issue is phony.
So, you're saying that if some entity in the private sector makes a direct cash payment to a specific public official in order to procur a lucrative contract for the provision of widgets to the government at three times their fair market value, it is "illegal bribery" but if the same private entity employs a lobbying firm to make a direct cash payment to the party boss who then deposits a portion of the payment into the campaign chest of the same public official as a laundered general political contribution, with the unwritten (perhaps, not even directly spoken) understanding that said public official will direct said contract to said private entity in exchange for said contribution, it is ....
...what?
Please answer the question, if you don't mind, because if this is not bribery I do not know what it is.