If the headlights, taillights, fog lights and brake lights are all functioning OK and only the turn-signals are malfunctioning, I’d bet the replacement bulb is the problem, or another turn-signal bulb has gone bad. Flip the signal lever both ways and check the signal lights on all four corners of the car. Typically, if the signal indicator light on the dash stays on there is a bad bulb(s), and if the fuse is bad the dash indicators won’t come on at all.
It might be that the front turn signal replacement bulb you’ve used is the wrong one (an easy enough mistake to make), a bad bulb from the factory (it does happen, once in a million) or the bulb was put in wrong (hard to do, but possible). Take out the one that you put in as a replacement and then see if the other side turn signal works now – it should.
Double check the application, look in the owners manual or look up the manual online.
I believe that the manual calls for a 3157NA, make sure that’s what you have.
You didn’t say why you replaced any of the bulbs to begin with, which would be helpful, but it’s possible that a rear turn-signal bulb was the cause of the turn-signal problem to begin with, or at least is the problem now.
As mentioned a ground could be at fault.
I’m not a 100% sure, but I think ’99 Olds 88s had gone to a lighting module and won’t have a conventional flasher.
Keith