Bugün hava sıcak, bu yüzden şort giyiyorum.

In Turkish, the verb to be is often omitted in the present tense when you are making a simple statement.

So:

• hava sıcak = the weather is hot
• literally: weather hot

This is completely normal in Turkish.
If you wanted to say it in the past, you would need an ending or a separate form:

• Hava sıcaktı. = The weather was hot.

So in the present, hava sıcak already naturally means the weather is hot.

These mean different things.

• hava sıcak = the weather is hot
• sıcak hava = hot weather / warm air

In hava sıcak, sıcak is the predicate: it tells you something about hava.

In sıcak hava, sıcak directly describes the noun hava, like an adjective before a noun.

So:

• Bugün hava sıcak. = Today, the weather is hot.
• Sıcak hava beni yoruyor. = Hot weather tires me out.

This is a very common distinction in Turkish.

Bu yüzden means because of this, for this reason, or more naturally in English, so / therefore / that's why.

In your sentence:

• Bugün hava sıcak, bu yüzden şort giyiyorum.
• Today the weather is hot, so I’m wearing shorts.

Breakdown:

• bu = this
• yüzden = because of / due to

Together, bu yüzden introduces a result.

Similar expressions:

• Bu yüzden geldim. = That’s why I came.
• Yağmur yağıyor, bu yüzden çıkmıyorum. = It’s raining, so I’m not going out.

Giyiyorum comes from the verb giymek, which means to wear or to put on, depending on context.

Here is the breakdown:

• giy- = verb stem
• -iyor- = present continuous marker
• -um = I

So:

• giyiyorum = I am wearing / I wear

In this sentence, the natural translation is I’m wearing.

A few related forms:

• giyiyorsun = you are wearing
• giyiyor = he/she/it is wearing
• giyiyoruz = we are wearing

Turkish often leaves out subject pronouns when they are already clear from the verb ending.

So:

• giyiyorum already means I am wearing
• the -um ending tells you the subject is I

Because of that, ben is unnecessary unless you want emphasis.

Compare:

• Şort giyiyorum. = I’m wearing shorts.
• Ben şort giyiyorum. = I’m the one wearing shorts / I am wearing shorts (more emphasis)

This is very common in Turkish.

In Turkish, şort is the normal word for shorts as an item of clothing, even though English uses a plural form.

So:

• şort giyiyorum = I’m wearing shorts

This is just a vocabulary difference between the two languages. Turkish treats it as a singular noun.

If you want to make it plural in Turkish, you can say:

• şortlar = shorts (plural)

But in many everyday situations, şort by itself is enough when talking about the clothing item in general.

In this sentence, it is most naturally I am wearing.

The -iyor form is often called the present continuous, but in real Turkish it can cover several meanings depending on context, including:

• something happening right now
• a current situation
• sometimes a habitual action

Here, because of Bugün (today) and the context about the weather, the meaning is clearly:

• I’m wearing shorts today

So giyiyorum here refers to a present action or current choice.

Yes, bugün can move, but putting it first is very natural because it sets the time frame right away.

• Bugün hava sıcak, bu yüzden şort giyiyorum.
  = Today the weather is hot, so I’m wearing shorts.

You could also hear:

• Hava bugün sıcak, bu yüzden şort giyiyorum.

That can be okay too, but it may place slightly more focus on today in contrast with other times.

Turkish word order is flexible, but the beginning of the sentence is often used for context such as time, place, or topic.

Turkish does not have a definite article like English the.

So hava can mean:

• weather
• the weather

The exact meaning depends on context.

In this sentence, hava sıcak is understood as the weather is hot, because that is the natural meaning in context.

Turkish also has no direct equivalent of English a/an in many cases, although bir can sometimes play that role.

The sentence is:

• Bugün hava sıcak, bu yüzden şort giyiyorum.

A rough breakdown is:

• Bugün = Today
• hava sıcak = the weather is hot
• bu yüzden = so / because of this
• şort giyiyorum = I’m wearing shorts

Turkish often prefers the verb near the end of a clause, and that happens here with giyiyorum.

So the second clause follows a very typical Turkish pattern:

• object/item + verb
• şort giyiyorum = shorts I-am-wearing

That final-verb tendency is one of the most important word-order patterns in Turkish.

It can mean both, depending on context.

• şort giyiyorum usually means I’m wearing shorts
• in another context, it can mean I’m putting on shorts

Turkish does not always separate those two ideas as clearly as English does.

If the context is about what someone has on, giymek is usually understood as to wear.
If the context is about the action of dressing, it may be understood as to put on.

In your sentence, because of the weather explanation, I’m wearing shorts is the best interpretation.