Bisiklet pompası dış lastiğin yanında.

In Turkish, present-tense sentences with a noun, adjective, or location often do not use a separate word for is.

So:

Bisiklet pompası dış lastiğin yanında.

naturally means:

The bicycle pump is next to the outer tire.

Turkish often leaves the present-tense copula unspoken in this kind of sentence. If you change the tense, it appears more clearly:

• Bisiklet pompası dış lastiğin yanındaydı. = The bicycle pump was next to the outer tire.
• Bisiklet pompası dış lastiğin yanında olacak. = The bicycle pump will be next to the outer tire.

Bisiklet pompası is a very common Turkish structure called an indefinite noun compound.

Pattern:

noun + noun-(s)I

So here:

• bisiklet = bicycle
• pompa = pump
• pompası = pump + compound ending

Together, bisiklet pompası means bicycle pump.

This does not necessarily mean literal ownership, like the bicycle’s pump. It often just means a pump for bicycles or a bicycle-type pump.

Compare:

• araba kapısı = car door
• mutfak masası = kitchen table
• diş fırçası = toothbrush

The ending -sı is the compound marker here. In this structure, the second noun takes a third-person possessive-looking ending:

• pompa → pompası

Because pompa ends in a vowel, Turkish inserts -s- before the suffix:

• pompa + ı would be awkward
• so it becomes pompa + sı → pompası

This is normal:

• kapı kolu = door handle
• meyve suyu = fruit juice
• çocuk odası = child’s room

Even though this looks like a possessive ending, in compounds it often just helps form the expression.

Because yanında usually works with a noun in the genitive form when it means next to / at the side of something.

So:

• lastik = tire
• lastiğin = of the tire / the tire’s

The full structure is like:

lastiğin yanında = at the tire’s side = next to the tire

This same pattern is very common with location words such as:

• üstünde = on top of
• altında = under
• önünde = in front of
• arkasında = behind
• yanında = beside / next to

Examples:

• Masanın üstünde = on the table
• Evin önünde = in front of the house
• Arabanın yanında = next to the car

Also note the spelling change:

• lastik
  • -in → lastiğin

The final k softens to ğ before the vowel-initial suffix.

Yanında comes from:

• yan = side
• -ı = its
• -nda = at/in/on

So literally it is something like:

at its side

That is why lastiğin yanında means at the side of the tire or more naturally next to the tire.

This is part of a very common Turkish pattern with relational nouns:

• yanı = its side
• yanında = at its side
• yanından = from beside it
• yanına = to its side

For example:

• Evin yanına git. = Go to the side of the house.
• Evin yanında bekle. = Wait next to the house.

Dış means outer / outside, and lastik means tire.

So dış lastik means outer tire.

In bicycle vocabulary, this is often contrasted with:

• iç lastik = inner tube
• dış lastik = outer tire

So this sentence is talking specifically about the tire on the outside, not the inner tube.

Turkish often puts the main predicate at the end of the sentence.

Here:

• Bisiklet pompası = the bicycle pump
• dış lastiğin yanında = next to the outer tire

So the order is:

subject + location

That is very natural in Turkish.

English and Turkish do not always order things the same way, but this sentence is actually fairly close to English:

The bicycle pump + is next to the outer tire

Turkish just leaves out the separate word is.

Not in this exact structure.

Bisiklet pompası dış lastiğin yanında.
means The bicycle pump is next to the outer tire.

If you want There is a bicycle pump next to the outer tire, then Turkish usually uses var and a different focus:

• Dış lastiğin yanında bisiklet pompası var.

So the difference is roughly:

• Bisiklet pompası dış lastiğin yanında. = The bicycle pump is next to the outer tire.
• Dış lastiğin yanında bisiklet pompası var. = There is a bicycle pump next to the outer tire.

The first talks about where the pump is.
The second introduces the existence of a pump there.

Yes. Yanında can mean next to / beside, but in some contexts it can also mean with someone / on someone / in someone’s possession.

For example:

• Anahtarlarım yanımda. = I have my keys with me.
• Telefonu yanında mı? = Does he/she have the phone with him/her?

But in your sentence:

Bisiklet pompası dış lastiğin yanında.

the meaning is clearly spatial: next to the outer tire.

It is pronounced roughly like las-tee-in with a soft transition, not a strong hard g sound.

The ğ in Turkish is often called soft g. In many words, it does not sound like an English g. Instead, it usually lengthens or smooths the vowel before it.

Why is it there?

• base word: lastik
• suffix added: -in
• before a vowel-initial suffix, final k often softens to ğ

So:

lastik → lastiğin

This kind of change is very common in Turkish.