The new "conversations" are a little different than the old "PMs". The old PMs were saved as individual messages, so if you wrote back and forth with someone 5 times, you'd have 10 individual messages. (the 5 you received and the 5 you sent)
But with the new "conversations" the whole back and forth is kept together, like, well... a conversation. Similar to texting on your phone. You see the whole history of the conversation all in one spot.
So you can no longer "delete" individual messages like with the old forum. Now that the "conversation" is all kept together you can't delete it because you'd also be deleting the other person's messages on their side.
But what you can do is "leave" the conversation. That will remove it from your inbox. But be aware it doesn't remove it from the other person's inbox, just like with the old forum, deleting your messages didn't delete the other person's copy.
However, there's really no limit to the number of conversations anymore, so you don't need to delete old ones. If you're just being OCD and like a cleaned out inbox, then go ahead.
But it's not necessary to free up space.
See screen shots below to "leave" conversations.
You can then select multiple conversations using the checkboxes on the right and leave them all.