No
Let's take this feature at a time...
Quick build
This is just flatly wrong. This class is a half-caster, they should not prioritize their casting stat. Glancing down below, this class would always want to prioritize Dex--they are restricted to Light Armor no access to the Mage Armor spell and they don't have Cantrips for steady damage output. So they'll want a high Dex for AC, and to make use of Ranged or Finesse weapons--Str is not a good primary for a class that is restricted to Light Armor.
Proficiencies
Light armor only? This pigeonholes this class into a Dex-build if they want anything resembling durability. They are not a dedicated caster--don't even have Cantrips.
Draconic bond.
Basically a Familiar that can attack at this point. However, unlike a Chain Warlock's familiar, the Pseudodragon does not consume one of your attacks in order to act--as-written, this is a complete secondary character.
Spellcasting
You have the spell range of a half-caster, but weirdly scaling efficiency. Without playtesting, it would be hard to determine how this pans out.
That said, your spell list is basically a grab-bag of the most useful spells in the game drawn from every class's list. Balanced fairly well by the fact that they are mostly (all?) Concentration spells
As an aside--you need to rename this feature. Classes that don't follow the standard rules for spellcasting call their magic feature something else--like a Warlock's Pact Magic.
Fighting Style
You get the same full range of fighting style options as a Fighter does. There is basically precedent for this with a Paladin.
ASI
Normal, whatever, moving on.
Draconic Growth
And here is where the balance starts its nose-dive. Assuming you take a Blue or Gold dragon, you just added a CR 3 companion. At a rough ballpark based on experience, a CR 3 monster is roughly the equivalent of a 4th to 5th level character in combat. So, at 5th level, you are basically controlling 2 characters of roughly similar combat effectiveness.
For comparison, the wolf of a 5th level Beast Master ranger has 1 less AC, 1/3 the HP, and does a fair bit less damage while attacking. And, most importantly A ranger must give up an attack of their own to let their wolf attack. And that's not including the dragon's breath weapon.
I would also note that about this point--your 'pet' is lining up to be as good at Persuasion as your party's social lead unless you have someone with Expertise in Social Skills
Improved Bond
Seems handy in a pinch...not a huge deal, moving on.
Draconic Tendencies
Again, not a huge deal...but I would note there's nothing in there about what to do if you already have the Survival and Nature skills
Draconic Growth
So, the wyrmlings usefulness has gradually been tapering off and keeping it alive has gotten hard. That's gone again.
Assuming you went for optimization and took a Gold Dragon...you are now a level 10 player with a CR 10 companion. Again, based on experience, I ballpark that a CR 10 monster is at least on-par with a level 14 PC. Now you have a pet dropping three (very powerful) attacks per round, as well as their breath weapon--and almost certainly has more hp than your party's Barbarian.
Extra attack
So...now your character is up to 5 effective attacks per round? 2 from themselves, 3 from their Dragon.
Riding Master
A fragment of the Mounted Combatant feat...moving on
Greater Bond
I can see this stacking in pretty terrifying ways. It will always work unless aimed at a creature with Evasion.
Increased Draconic Tendencies
Fluff, moving on
Draconic Growth
Ah-hahahahahahahahahahahahahahahaha
Why are you still in a party? You have a CR 17 companion (significantly more powerful than a level 20 PC) that has Legendary Actions and Legendary Resistance. Other classes get a 4th Attack or regenerating class resources--but you? You get a pet legendary monster.
Subclasses
Weapon Master
Level 3
Seems niche and kinda weird..whatever
Level 7
Eldritch Knights don't get this until level 18
Level 18
Stronger than the Mounted Combatant feat, and that's some pretty serious damage...
Spell Master
Level 3
Okay, now you're almost a half-caster. Still really inefficient
Level 7
The lack of a range on this is questionable--otherwise this doesn't seem too out of place
Level 18
Again, pretty potent, but not nearly as broken as your pet flying apocalypse.
Summary
You have a boss monster as a free-acting companion. There is no possible way this is balanced against the core classes unless the storyline gives them some way to acquire creatures of similar power. Which you could do, of course--but that's not what you asked.
Ultimately, the character themselves is quite lackluster--with underpowered spellcasting, no extra attack until level 11, and being stuck in light armor.
However, they have a boss monster as a pet. The way gameplay with this class would go is that it'd be kinda meh until level 5--then you'd be the strongest character in the party by far. That would then taper off as enemies caught up with your dragon, though you're still controlling 2 full-fledged independent characters--til you had another massive surge in power at level 10 that would largely carry through until max level--with another (frankly obscene) power surge at level 20.
Best Answer
The class you're looking for is the Dragonfire Adept (Dragon Magic) which provides a very respectable breath weapon (especially at high levels) "spellcasting" and some alternate forms.
You'll want to read the dragonfire adept handbook, but possible invocations involve frightful presence, interesting shapechanging, dispelling at will (which is sometimes better than SR) and all sorts of fun with dragonbreath. To save time, read through the builds here until you find one that fits your intentions.
If you want to be huge and tough:
this provides significant cleric casting which nets you divine power (HD = BAB and a strength boost) and Righteous Might, which is a much better way of going up a size category. If you really want to tell puny mortals to cower, be a half-minotaur (LA +1 Dragon 313) or half-ogre (LA +2 dragon 313), will provide. Or start out as an anthropomorphic huge viper snake and paint your scales gold.
Instead, I'll recommend an alternate route. Since you're playing a high-level game, take cleric for your last levels, and make sure you have resources for divine metamagic and persist spell. Every day, cast divine power with persist spell and use miracle to persist giant size (wu-jen 7, castable without XP cost via miracle), Spell Resistance (It'll not be all that significant, but you'll have it), Spell Turning, Girallon's Blessing (For the claw, claw, claw, claw, rake series), and the other suite of spells. You'll be colossal and dragon-magical without needing to invest levels in a race.
Make sure your cleric has access to the hunger domain, which gives you a bite attack, and the domain spell "bite of the king" which gives you swallow whole. Unfortunately, there are no spells worth the cost that give you a tailslap or wing attacks, but your Claw Claw Claw Claw Rake Bite Swallow Whole attack sequence will simply have to make up for them.