Has this happened to anyone else? After doing two-three bounties neither the steward nor innkeepers in Whiterun offer me any more of them. Companions too, I finished the questline and I can only get jobs from Vilkas - and none of them has been in Whiterun.

I still have a lot of undiscovered dungeons in the hold, and I never even visited a single Giant Camp. So it's not like there's no more work to be done...

Each hold will have bounties. You will need to try different inn keepers and explore a little for this. Also try talking to random people. This can create more quest. I have 118 hours in and have yet to do any of the main quest. All I have done is side quests and radiant quests. Also exploring can reveal quest from books or notes.

I may be mistaken, but I had the impresion that the bounties should be unlimited as far as you report to the jarl after completing each of them.

you have to wait a certain time, just try again later

Sometimes you get the same task several times. I seems unlimited

You might also have a task to clear. I didn't get new messengers with dragon wall hints when I had one on the map. I also think that hints added to maps need cleared before you get new hints to keep from cluttering. So if you see a black icon, visit!