Make it legal, then. Regulate it, call it a fee and problem solved.
No. The moment bribe is legalized, it stops being a bribe. That is, the officials get a kick out of this whole shady operation. If we deny them this pleasure, they will find other ways.
As it stands today, there is no accountability in the whole system. An official can sit on a file for years with no repercussions. You can drag them to court, but sooner than later you will understand that this is a battle you can never win. The official has the power and money of the government behind him while you are totally alone. Most importantly, the official has time while you do not.
The rates are fixed. The collection mechanisms are solid. Everyone, from the peon to the minister, gets a cut.