Ever considered how musicians play etudes and scales to work on certain aspects of their technique? Or, how athletes run drills to improve their performance in particular areas? The same can apply to writing poetry- you can target a certain aspect of writing, apply relevant rules to the act of writing, and improve that particular thing.
In this case, we'll look at ideas and rules for improving concision- that is, "exercising brevity" (oh ho!). If you tend to write really long poems, whether they be free verse, formal verse, or something in between, then obvious rule is obvious- limit the number of lines you have! For example, a sonnet accomplishes this. Take something that you could (and would) probably write 30, 40, 50 lines about, and write 14. That's it. Doesn't necessarily have to be a classical sonnet (it can be modern), but you only have so many words, so use them more effectively.
From the basic rule, you can also add other rules or try tougher forms to make the process more technically challenging (especially if you tend to write free verse). Villanelles, for example, are challenging- sure, you have 19 lines, but you also have a refrain that has to repeat multiple times, and to write a good villanelle, that refrain has to be written in a way that lends itself to multiple meanings, which is something that, in general, helps with concision. Or, for additional rules, you can cut down further on the lines (from a sonnet benchmark), impose a rhyme scheme, take a longer work (like a short story or fairy tale) and condense it while still retaining most (or all) of the force and meaning, etc.. Here's an example of how a fusion of these suggestions works:
The Thieves and the Cock
Some thieves broke into a house and found nothing worth taking except a cock, which they seized and carried off with them. When they were preparing their supper, one of them caught up the cock, and was about to wring his neck, and cried out for mercy and said, “Pray do not kill me. You will find me a most useful bird, for I rouse honest men to work in the morning by my crowing.” But the thief replied with some heat, “Yes, I know you do, making it still harder for us to get a livelihood. Into the pot you go!”
(Translated by V.S. Vernon Jones)
The Thieves and the Cock
The silent party made their break
And stole a meal. Then cried the cock:
“Please let me go! I’ll help- I wake
Good men each day!” Of this he talked
Till a thief replied and grabbed his bill:
“Yesyes, you silly bird, I know-
you make our jobs even harder still.
Now— into the pot you go!”
Lessee- cutting down on lines? Check- only 8. Rhyme scheme? Check- abab, very standard, and appropriate for this kind of material. Taking a work and condensing it? Check- 107 vs. 62 words, and every important element of the story is still there. Is this a really great poem? No, but it's definitely solid, and it does stand on its own, apart from the prose version. And, the lessons learned by working within such restraints can easily translate over to longer and more complicated works, which is essential.
A side note, since I've talked quite a bit about imposing rules on writing to improve: if you tend to always write with formal rhyme schemes, or always write really short poems, the opposite rules can also apply. Make yourself write free verse, and work on expanding ideas and adding extra layers and nuances to give the words meaning and maintain interest. But, the more common problem (as I noted at the beginning of Part I) is overwriting, which I chose to mainly address.
Although certain things are very difficult to practice, such as the development of interesting and fresh ideas, other things can be systematically improved upon, such as the presentation and execution of ideas. And that, in itself, is worth something. A piece with an interesting idea, but poor execution of that idea, is mediocre at best- and who doesn't want to strive for something beyond mediocrity?