From personal experience: I don't bribe policemen, I donate to the police union or contribute money towards some officer's or family member's health or college fund. I don't bribe school administration, I go to a few Booster Club events a year and throw money at whatever their latest stupid fund is. I don't bribe teachers, I show up on the first day of school with enough science lab kits and other school supplies for a whole class. I don't bribe my doctors, I contribute money to the nonprofit clinics they are partnered with, which then pays part of the doctor's annual salary directly so that they can focus more on the clinic.
There is far more indirection and the "return on investment" is certainly far less than what I saw when living in Russia, for example, but that's just lipstick on a pig because at the end of the day, it works. When my kid and his friends were caught with drugs (not cannabis), I got to the station less than 5 minutes after they did and was able to get them out by the next day without even seeing a judge (the evidence was "misplaced" on the way to lockup so the prosecutor felt it wasn't worth his time). When he got into a fight with a close friend, I managed to get both of them off with an in-school suspension instead of an expulsion that would go on their permanent records. Never once did I bring up the donations or threaten to cut them off - my rhetoric in each situation basically boiled down to "boys will be boys but I'll make sure they know what it means to be punished" but the deference I received was far more than the average person would, even when getting special treatment for someone else's kid.
For example, when some party gets shut down by the cops the kids of the grocery store owner who donates all the food for some annual police/fire/vfw bbq don't have to worry about getting misdemeanors. When it's time to determine who gets what contracts the company that generously sponsored the boy scout troop that the son of whoever is making the decision will get special consideration (all the blue collar small businesses don't sponsor local stuff because it has a negative ROI).
For example, I live in LA and the 90s were very tough for police all across the state after the Rodney King riots. I was buddies with one of the guys heading up reform in the LAPD after the riots so I "donated" a thousand bucks to have him come out for a weekend and teach the basics of community policing to my underfunded local PD - something that I felt was important anyway because of the wealth inequality in my town. The department didn't have it in their budget to hire him themselves but they did have state money for overtime for police training so not only did every officer get training but they got paid extra to do it. Since my name was dropped multiple times as the sponsor of the training, all of the officers in the department knew who I was and associated my name with someone who supports the department in concrete ways (even if years later, they couldn't remember why!). It also didn't hurt that the training opened their eyes to new strategies at a time when tensions between police and civilians were boiling over.
Also, you can actually do this by donating a little bit to a lot of things but it has to be strategic. For example, a few members of the Booster Club and I have an ongoing scholarship at my local high school where we throw in $250 per donor per student for textbooks for anyone whose parents didn't go to college. Since my town is very wealthy, the number of students who qualify for this scholarship is minimal (a few thousand a year split between 4 people) but it is a guaranteed scholarship to anyone who qualifies so every teacher and administrator knows about the scholarship and who the donors are. Since we are explicit about the money going towards textbooks, it also sounds like a lot more than it really is because people think of the scholarship in the narrow context of textbook costs instead of overall graduation cost.