Whenever I as GM am dealing with multi-PC/NPC conversations, I first remind my players that I am a single-thread processor and can therefore only handle one conversation at a time, then proceed to deal with them one at a time as appropriate for the configuration of speakers.
1. One-to-Many Conversations
The easiest way to handle this situation is to enforce multiple one-to-many conversations (as opposed to a single many-to-many conversation). This allows multiple PCs who all want to talk to the same NPC to do so as a group; then the PCs who want to talk to another NPC to do so. For example, if John, Jade, and Rose all want to talk to the NPC Jack, I will hold that conversation all at once. Dave, Jane, and Roxy, who want to talk to the NPC Mayor, wait their turn and have their conversation separately after I've finished with the first one.
I also do the reverse: John wants to talk privately to both the Mayor and Jack, so I handle that conversation all in one go, while the other PCs wait their turn for their own conversations.
This is probably the ideal way to handle a bar scene like the one you describe, since bar chatter is typically several one-on-one or one-to-many conversations. While it feels a little artificial, since in-game the conversations would all be taking place at once, it allows you as the GM to focus on each conversation and give the involved PCs your full attention.
2. Many-to-Many Conversations
The best way to handle many-to-many conversations as a GM is to cheat and turn them into one-to-many conversations. When multiple PCs are talking to a group of NPCs, pick one NPC to be the group's "spokesperson", who does most or all of the talking. Occasionally, where plot-relevant, another NPC might interject a few times, but for the most part, this allows a many-to-many scene to actually be a one-to-many conversation.
For example, if the whole party wants to talk to a rival party, I will pick the most talkative of the rivals, and have them do almost all the talking. The rest of the rivals are present, but silent in the background. If I want to illustrate that there's an ideological rift in the rival party, then I might have a second NPC jump in while the spokesperson is talking, but that dialogue is handled as an NPC-to-NPC dialogue. If the party wishes to speak to the second NPC (perhaps because they think she'll be more sympathetic to their arguments), then the first NPC falls silent and the second NPC becomes the spokesperson. Again, this enforces the rule of one-to-many conversations and prevents the GM from being overwhelmed.
3. Multiple Important NPCs
Having multiple important NPCs in a scene means the GM needs to make a choice between making the scene a "cutscene" wherein the NPCs converse with each other, or making the scene a conversation between the NPCs and the PCs. In other words, is the point of the scene to convey a discussion between the NPCs, or is it to allow the PCs to talk to the NPCs?
If it's the former, then the scene can be a brief "cutscene" where the NPCs talk to each other and the players listen. Then, after the NPCs' different viewpoints have been showcased by the cutscene, the PCs can pick one of the NPCs to have a one-to-many conversation with, as described in #1 above.
If it's the latter, then the scene can make use of the spokesperson technique. Even groups with multiple powerful players tend to have a single "speaker", even if that person isn't the leader in any sense of the word. Not everyone is equally talkative, and the most talkative NPC will dominate the conversation with the PCs, leading to (again) a one-to-many conversation with a few interjections by the other NPCs, as described in #2.
As an added bonus, you can use subtle cues in this sort of scenario to showcase group politics. For example, if the spokesperson is respected, the rest of the group falling silent and allowing her to speak for them will indicate this. Alternately, you can note that although one person is doing all the talking, other NPCs are rolling their eyes, shaking their heads, or exchanging meaningful glances behind his back. This indicates that the spokesperson might be the most talkative, but not the most respected, member of the group - and may give the PCs a clue that they need to draw out a different person as the speaker.
Again, the key is to remember that you the GM are a single-threaded processor. No matter what, you cannot effectively have three or four or five conversations as different NPCs with different PCs at once. The best you can do is to cheat in ways that allow you to only ever have one effective conversation going at once.
Remind your players that you are one person and can only have one conversation at a time. Enforce one-to-many conversations by handling one conversation group at a time, and/or by using a spokesman for multiple NPCs.