İddianın sağlam bir dayanağı yok.

Literally, it is something like:

• iddianın = of the claim
• sağlam = solid / sound
• bir = a
• dayanağı = its basis / support
• yok = there is not / does not exist

So the sentence is built as:

X-in Y-si yok = X has no Y

That is why the natural English meaning is The claim has no solid basis.

Because Turkish is using a possessive construction.

In English, we say the claim has a basis.
In Turkish, this is usually expressed more like:

the claim's basis exists / does not exist

So:

• iddia = claim
• iddianın = of the claim / the claim's

The ending -ın is the genitive ending, marking iddia as the possessor.

Because it has a 3rd person possessive ending.

The base noun is:

• dayanak = basis, support

But in this sentence it means its basis or the claim's basis, so Turkish adds the possessive suffix:

• dayanak + ı → dayanağı

This is part of the standard Turkish pattern:

• iddianın dayanağı = the claim's basis

So both parts work together:

• iddianın = the possessor
• dayanağı = the possessed thing

This is a common sound/spelling change in Turkish.

When some nouns ending in k take a vowel-initial suffix, the k often softens:

• dayanak → dayanağı
• kulak → kulağı
• renk → rengi

So here:

• dayanak + ı does not stay dayanakı
• it becomes dayanağı

Also, Turkish ğ is usually not pronounced like a hard English g. It often lengthens the previous vowel or creates a smooth transition.

Here sağlam means sound, solid, well-supported, strong.

It does not mean physically solid in this sentence. It is being used figuratively.

So:

• sağlam bir dayanak = a solid/sound basis
• sağlam bir kanıt = solid evidence

This is a very common figurative use in Turkish.

Bir is the normal word for a/an with an indefinite singular noun.

So:

• sağlam bir dayanak = a solid basis

When you add yok, the whole sentence becomes negative:

• sağlam bir dayanağı yok = it has no solid basis

So even though bir literally means a, the natural English translation with yok is often no or any.

There is no separate verb like English is or has in the present tense here. That is normal in Turkish.

The word yok acts as the predicate and means there is not / does not exist.

So instead of saying:

• The claim does not have a solid basis

Turkish says something closer to:

• The claim's solid basis does not exist

This is a very common Turkish pattern.

Because yok and değil do different jobs.

• yok = non-existence / absence
• değil = not / is not

So:

• İddianın sağlam bir dayanağı yok = The claim has no solid basis
• İddianın dayanağı sağlam değil = The claim's basis is not solid

These are similar, but not identical.

The first says there is no solid basis.
The second suggests there is some basis, but it is weak or not convincing.

Not exactly.

• İddianın sağlam bir dayanağı yok = The claim has no solid basis
• İddianın dayanağı sağlam değil = The claim's basis is not solid

The difference is subtle but important:

1. ... yok focuses on absence

   • there is no solid supporting basis

2. ... sağlam değil focuses on quality

   • there is a basis, but it is not strong

So the original sentence is a little stronger in meaning.

Yes, very common.

Turkish often expresses have with var and yok instead of a separate verb meaning to have.

Examples:

• Arabam var. = I have a car.
• Arabam yok. = I don't have a car.
• Bu planın mantığı yok. = This plan has no logic.
• İddianın sağlam bir dayanağı yok. = The claim has no solid basis.

So this sentence uses one of the most important everyday Turkish patterns.