I´m learning Turkish now through Rosetta Stone and unfortunately they don´t explain new grammar they introduce.  

People are saying "Ne dedin/dedi?" which i understand perfectly but the grammatical responses are confusing me. 

For example "Daha büyük bir bavula ihtiyacim olduğunu söyledim." I firstly don´t understand that suffix, and also do not understand the use of olmak in this sentence as I´ve found it is used quite sparingly, only when absolutely necessary and in this sentence the verb to be doesn´t seem necessary? One could just use var. 

Other examples are, 

Bu bitkinin yağmur ormanlarında yetiştiğini söyledi. 

Parkın batıda olduğunu söyledi. 

Thank you! I apologize I feel like I ask questions on here all the time. 

I don´t fully understand it myself. But consider:

You are incorrect. Hatalısın.

I believe you are incorrect. Hatalı olduğuna inanıyorum.

In English ´you are incorrect´ is the same in both (to be). But in Turkish the grammar changes. 

Sort of similar to the meyi when you have 2 verbs.

The suffix in olduğunu and yetiştiğini is -dik. Here is a good ref.:

http://www.turkishlanguage.co.uk/objectpart.htm

There is no uğunu/iğini suffix. There are three suffixes: 

-dik (´that´, see here: http://www.turkishclass.com/turkish_lesson_297)

-un (you/your)

-u (accusative suffix)

Olduğunu= that you are

or four: 

-dik

-u (he/his)

-n (buffer letter)

-u

Hi Emily, because I´m not certain of how much Turkish grammar you know, I´ll try to answer it as thoroughly as I can, but be aware that I´m also a learner.

Demek is used for directly quoting what is said, and usually uses inverted commas to quote the words spoken.

Söylemek is used for indirectly repeating what was said, and has 3 basic forms. It always uses the accusative object suffix i/ı/ü/u as the final suffix before a version of the verb söylemek at the end of the sentence.  

The first form is when imperative tense is used (instructions or commands) in reported speech. It uses the verb + me/ma + person suffix + accusative suffix

Example:

Emily bana "Bulaşıkları çabuk yıka" dedi.

Emily said "Clean the dishes quickly" to me.

In reported speech this becomes

Emily bana bulaşıkları çabuk yıkamamı söyledi.

Yıka + ma + m + ı

Emily told me to clean the dishes quickly.

The second form is used when past or present tense in used in the reported speech. It uses the verb + dik/duk + personal suffix + accusative suffix.

Whenever there is no verb used in the reported speech, ie when using "is/are", the verb olmak is utilised in the reported speech.

Examples

Annem bana "hava çok sıcak" dedi.

My mother said to me "The weather is very hot".

Annem bana havanın çok sıcak olduğunu söyledi.

My mother told me the weather is very hot.

Babam bana "Param var" dedi.

My father said to me "I have money".

Babam bana parası olduğunu söyledi.

My father told me he has money.

Babam bana "Param yok" dedi.

My father said "I´ve no money".

Babam bana parası olmadığını söyledi.

My father told me he hasn´t got money.

The third form is when future tense is used in the reported speech.

It uses verb + ecek/acak + personal suffix + accusative.

Example

Ben ona "Yarın işe erken gideceğim." dedim.

I said to her "Tomorrow I will go to work early".

Ben ona yarın işe erken gideceğimi söyledim.

I told her that tomorrow I´ll go to work early.

I hope this explanation is useful to you. 

Exactly, as maybe an easy reminder: when you use ´-dik´ you can use ´that´.

Also a question: is there also a 4. and 5. form, which are abil/ebil and malı/meli? (Or do they fall under the 3. form?)

Or any other?

That´s what this forum is for honey

For reported speech using meli/malı (suffix for must) gerektiğini is used.

Example

Arkadaşım bana "Eve gitmeliyim" dedi.

My friend said to me "I must go home".

Arkadaşım bana eve gitmesi gerektiğini söyledi.

My friend told me that he must go home.

Note gerektiğini is always the same, and only gitmek changes its form

I = gitmem, you = gitmen, he/she = gitmesi, we = gitmemiz, you (formal & plural) = gitmeniz, they = gitmeleri

For the abil/ebil suffix, it depends on the tense, and whether it is used in a question.

Elena bana "Türkçe okuyabiliyorum" dedi.

Elena said "I can read Turkish".

Elena bana Türkçe okuyabildiğini söyledi.

Elena told me that she can read Turkish.

Examples with questions use the verb + ip/up followed by verb + a/e + ma/me (negative) + y (buffer) + ecek/acak + personal suffix + i/ı

Öğetmenime "Bir soru sorabilir miyim?" diye sordum.

I asked my teacher "Can I ask a question?"

Ben öğretmenime bir soru sorup soramayacağımı sordum.

I asked my teacher whether (or not) I could ask a question. 

Please correct if any Turks see errors.

Excelleeeeeent thank you!

I hope I never have to tell someone that I asked someone if I could ask them a question ..... 

It´s like a tongue twister 

soru sorup soramayacağımı sordum.

Being an aglutinating language Turkish can sometimes sound really funny to foreigners, I guess.

Thank you all sooooo much!!! Super helpful!

So the sentence 

Bizim takımın kaybedeceğini mi düşünüyorsun? 

You thought our team would win? 

I believe it´s: Do you think (or - are you thinking) our team will lose?