I'll give it a try, tell me if its not what you're looking for.
Let's look at Pokemon that check and counter Azumarill.
- Mega Venusaur
- Ferrothorn
- Skarmory
These three are probably the most common and most effective counters to your Azumarill.
Mega Venusaur is the biggest threat here. It resists both of Azumarill's STABs, and can take any coverage moves because of its natural bulk, retaliating with super-effective STAB moves.
Ferrothorn also resists both STABs, and can setup entry hazards and Leech Seed.
Skarmory can also setup entry hazards while Roosting of any damage from Waterfall, and possibly using Brave Bird or Drill Peck to slowly kill your Azumarill.
There are other Pokemon that your Azumarill may have problems with because it doesn't have maxed out Attack, but these are the biggest threats.
OK, now let's look at checks and counters to these Pokemon:
Talonflame @ Sharp Beak
Ability: Gale Wings
EVs: 252 Atk/ 4 SpD/ 252 Spe
Jolly Nature
- Brave Bird
- Flare Blitz
- Roost
- Swords Dance
Sharp Beak allows Talonflame to have a Jolly nature while still being able to land a guaranteed 2HKO on Mega Venusaur with Brave Bird.
Ferrothorn and Skarmory are very Defensive Pokemon, however STAB Flare Blitz will still scare them out Talonflame's big weakness to Stealth Rock can be an issue though, especially when facing these hazard setters, as it makes it hard for Talonflame to be switched in and out, so I reccomend having a Rapid Spinner on your team.
Azumarill and Talonflame synergise well, as they check eachother's counters, however they do share a weakness to Electric, but that cannot be helped.
Magnezone @ Choice Specs
Ability: Magnet Pull
EVs: 172 HP/ 252 SpA/ 84 Spe
Modest Nature
- Hidden Power Fire
- Thunderbolt
- Flash Cannon
- Volt Switch
I know you asked for fast Pokemon, but Magnezone is very good at dealing with Azumarill's counters, which generally consist of bulky Grass types and Steel types.
Magnet Pull allows Magnezone to trap the likes of Ferrothorn and Skarmory, dispatching them with Hidden Power Fire, a OHKO and a 2HKO on Ferrothorn and Skarmory, respectively. It also takes care of Scizor, which might be difficult for your specific Azumarill to deal with.
It resists both of Mega Venusaur's STAB, but it may be Leech Seed stalled.
If its a problem, then you could also use this set:
Magnezone @ Leftovers
Ability: Magnet Pull
EVs: 84 HP/ 252 SpA/ 168 Spe
Modest Nature
- Hidden Power Fire
- Thunderbolt
- Flash Cannon
- Substitute
This set allows Magnezone to be impervious to Mega Venusaur's Leech Seeds, but at the cost of Choice Specs, which allowed you to OHKO Ferrothorn. the 168 Speed EVs are to allow Magnezone to out speed non-invested Mega Venusaur.
Latios @ Life Orb
Ability: Levitate
EVs: 252 SpA/ 4 SpD/ 252 Spe
Timid Nature
- Draco Meteor
- Psychic/ Psyshock
- Hidden Power Fire
- Roost
A fast one, but it is out run by other base 110 Speed Pokemon. Hidden Power Fire allows it to deal with Ferrothorn and Scizor, but the cost is 1 Speed IV, but it's perfectly good at checking Azumarills counters thanks to its Psychic typing
Skarmory @ Rocky Helmet
Ability: Sturdy
EVs: 252 HP/ 152 Def/ 100 Spe
Bold Nature
- Taunt
- Stealth Rock
- Spikes/ Drill Peck/ Whirlwind
- Roost
Yes, ironically, Skarmory can be a good support for Azumarill.
The most important move here is Taunt, because this, paired with Skarmory's Steel typing and high Defense, shuts down most of Azumarill's checks.
The problem is that offensive Mega Venusaur probably has either Hidden Power Fire or Hidden Power Ground, so it can be risky.
Heatran @ Air Balloon
Ability: Flash Fire
EVs: 4 Def/ 252 SpA / 252 Spe
Timid Nature
- Fire Blast
- Earth Power
- Flash Cannon
- Stone Edge
Heatran walls Mega Venusaur, unless it has Earthquake, but it's pretty uncommon. The reason for offensive Heatran is because due to Thick Fat the Defensive variant finds it hard to deal decent damage to Mega Venusaur, and if they stay and Leech Seed M-Venusaur could stall if it had Synthesis.
There are some more, such as Fire type Pokemon with powerful STAB moves, but I won't do them all.
Hope this helped
