step1: Accept the quest from Jarl Igmund to return Hrolfdir's shield. Also accept the bounty quest from the steward Raerek to kill the leader of Dragon Bridge Overlook (this may well work with any random bounty quest) or have 1 or both quests completed already but not turned in yet.
I completed both quests concurrently before returning to the Underkeep. When I returned, I found that I had the option to either return the bounty quest to the Jarl or to his steward (even though the quest states that I'm to return to Raerek for the bounty). When talking to either the Jarl or Raerek, if I turn in the bounty quest first, before turning in the shield to the Jarl, there are no further issues.
BUT, if I turn in the shield to the Jarl first, I'll then have the following 2 dialogue options with him:
a) "My Jarl, is there anything else you need?"
b.) The leader of Dragon Bridge Outlook is dead, I'm here for the bounty."
If I choose option a) first, he'll grant me the ability to buy a house from his steward, and then I can safely turn in the bounty quest with him safely after that and have the ability to buy the house from the steward. However, if I choose option b.) first, he will give me the 100 gold for the bounty, and option a) will disappear from the dialogue and I will not be able to become a Thane or even purchase a house.
Alternatively, if I only choose option a) from the Jarl, and then turn to his steward Raerek, I'll have the following dialogue options:
a) "The Leader of the Dragon Bridge Overlook is dead, I'm here for the bounty."
b.) "I'd like to purchase a house in the city. (8000 gold)"
If I choose option a) first and collect the bounty, the option for the house disappears, and I cannot get him to offer me that option again! If I choose to buy the house first, and then choose a) to turn in the bounty, everything is ok.
I will copy/paste this bug to Wiki Skyrim bugs, but if any of you folks can replicate this bug using a different bounty quest or even trying it with a different Jarl/Steward in another province would be cool. I've unfortunately bought all the previous houses and don't have any old save files to try that on.