Bu tişörtün bedeni bana büyük.

The ending -ün is the genitive ending, which often means of.

So:

• tişört = T-shirt
• tişörtün = of the T-shirt / the T-shirt’s

In Bu tişörtün bedeni..., Turkish is literally saying something like:

• this T-shirt’s size...

So tişörtün shows that bedeni belongs to or is associated with this T-shirt.

Because Turkish uses a special pattern for noun phrases like the size of the T-shirt.

When the first noun is in the genitive (tişörtün), the second noun usually takes a possessive ending:

• tişörtün bedeni = the T-shirt’s size

Here:

• beden = size
• bedeni = its size / the size of it

This is a very common Turkish structure:

• evin kapısı = the door of the house
• arabanın rengi = the color of the car
• tişörtün bedeni = the size of the T-shirt

In clothing contexts, beden means size.

So here bedeni is talking about the T-shirt’s size, not its physical body or shape.

For example:

• Bu gömleğin bedeni küçük. = The size of this shirt is small.
• Kaç beden giyiyorsun? = What size do you wear?

In everyday English, we would usually just say This T-shirt is too big for me, but Turkish can express that through its size.

Bana means to me / for me.

It is the dative form of ben (I):

• ben = I
• bana = to me / for me

In this sentence, bana büyük means:

• big for me
• more naturally: too big for me

Turkish often uses the dative with adjectives like this when talking about how something fits or suits someone:

• Bana küçük. = It’s too small for me.
• Bana uygun. = It’s suitable for me.

In clothing and fit contexts, bana büyük usually means too big for me, even though the word çok or a separate word for too is not used.

So:

• Bu tişört bana büyük.
  naturally means This T-shirt is too big for me.

This is just how Turkish commonly expresses fit.

If you want to make the idea stronger, you could say:

• Bana çok büyük. = It’s much too big for me.

But even without çok, the sense of too big is often understood.

In Turkish, the verb to be is often not stated in the present tense, especially in the 3rd person.

So:

• Bu tişörtün bedeni bana büyük.

literally looks like:

• This T-shirt’s size to me big.

But it means:

• This T-shirt’s size is big for me.

This is completely normal in Turkish.

You may sometimes see a formal ending like -dir, but in everyday speech it is usually omitted:

• Bu tişörtün bedeni bana büyüktür.
  This is grammatical, but it sounds more formal or marked.

Yes, absolutely. In fact, Bu tişört bana büyük is very natural and probably more common in everyday speech.

Compare:

• Bu tişört bana büyük. = This T-shirt is too big for me.
• Bu tişörtün bedeni bana büyük. = This T-shirt’s size is too big for me.

The version with bedeni is a bit more explicit because it specifically mentions the size.

So the sentence in your example is correct, but in casual conversation many speakers would simply say:

• Bu tişört bana büyük.

Turkish usually puts the main predicate at the end, so büyük comes last.

The sentence breaks down like this:

• Bu tişörtün bedeni = this T-shirt’s size
• bana = for me
• büyük = big

So the structure is roughly:

• Subject + for me + adjective

This is normal Turkish word order.

A more literal English-style order such as Bu tişörtün bedeni büyük bana would sound wrong or at least very unnatural.

Yes, but bana büyük is more natural for clothing fit.

Compare:

• Bana büyük. = It’s too big for me.
  This is the most idiomatic choice for fit/size.
• Benim için büyük. = It’s big for me.
  This is understandable, but less natural in this context.

So when talking about clothes, shoes, rings, and similar things, learners should strongly prefer:

• bana büyük
• bana küçük

No, bu stays bu.

In Turkish, demonstratives like bu (this) usually do not take the genitive ending when they directly modify a noun in a phrase like this. The noun itself takes the needed ending:

• bu tişörtün bedeni = the size of this T-shirt

So the genitive marking goes on tişört:

• tişört → tişörtün

not on bu.

That is why bu tişörtün bedeni is correct.

It sounds neutral and natural, though slightly more explicit than everyday casual speech.

• Bu tişörtün bedeni bana büyük.
  neutral, clear, a bit more specific
• Bu tişört bana büyük.
  more everyday and conversational

So if you are speaking casually in a shop, many people would probably say:

• Bu bana büyük.
• Bu tişört bana büyük.

But the full sentence with bedeni is still perfectly normal and useful.