Tidehunter, TANK or SUPPORT?

Is it better to play Tidehunter as a full support or as a tank initiator?

Tide can be played as both for he doesn't need items to be tanky as his high STR gain, natural damage block and damage reduction give him all the tankiness he needs. You can build him full tank bot only a heart of terrasque will suffice. Initiators are usually tanky or hard to kill. As for support, tho he is able to rightclick effectively his full potential arise as an offensive support.

He's already tanky, a Shiva's Guard, Heart of Tarrasque or even Pipe of Insight will make him durable enough.

Also, he needs a dagger to be an effective initiator.

You'd be surprised at how tanky Tidehunter is with just a bracer or two.

Yeah he's a semi-support - i.e. a support who wants a few items to maximise his impact, e.g. a Blink Dagger. (pair of Arcane Boots and a Bracer or two is nice as well)

Personally I'd rather get a Refresher Orb on him than a Heart of Tarrasque, but it does depend what your team needs from you.

Is it normal to be under leveled and under farmed as Tidehunter?
(if the ganks failed)

or am i doing bad ?

Your early farm is very dependent on whether you can get assists, stacks/pulls and whether the team is getting towers or not. As Tide you should be getting a courier but not warding (as a general rule), so you should have some farm if not a lot, enough to get Arcane Boots and a Bracer anyway. If a bit of midgrame quiet breaks out, try and grab a few waves of creeps if your carry isn't about - you can farm pretty fast with Anchor Smash and that should be enough to get you a Blink Dagger, along with some team fight and gank assists.

While you want to try and let your carry level a little bit ahead of you in lane, it makes a massive difference to the team when you get your ultimate, so it's ok to gobble up some xp (just not last hits).

I generally play Tidehunter with a lot of emphasis on support. I feel like everytime I try to get a Heart of Tarrasque it takes 25 minutes.

First, I just want to note that a tank is not really a role in Dota 2, as per say, a support and initiator.
Tide CAN be played as a 5 (hard support => if doing bad, under leveled and under farmed), but there are better options for a 5. Playing him as 4 probably best, once you your dagger, start building support items (mek if no one has it, drums, pipe, etc). Because all you really need to do is get a good ult off, and your team should do the rest.