Why not use t' instead of tu?

In general, grammar questions on ‘why’ are answered ‘because that’s the way it is’. Doesn’t mean there isn’t some reason and structure, but that there may be no real sense or logic underpinning them and exceptions abound.

Elision is an obligatory contraction of 2 words in French. It occurs in speech and writing, whether casual or formal. The following link will help understand the predictable, formal elisions. 

https://www.lawlessfrench.com/pronunciation/elision/

I think you may also be confusing ‘elision’ and ‘liaison’ in your example as there is no contraction written, and the apostrophe isn’t correct and doesn’t indicate a removed letter ? Liaison is a matter of pronunciation, not of spelling or grammar. It has more complex 'rules’ and will take time and practice to come to grips with. Pronouncing "nous aimons" as 'nou (z)aimons' is a liaison. See introductory link.

https://www.lawlessfrench.com/pronunciation/liaisons/ 

As if it is not already difficult enough for non-native speakers, in informal speech (and reflected in informal transcriptions) there are many non-standard ‘contractions’ which are not recognised elisions - such as “tu as” being pronounced and written “ t’as “. 

There is also the renowned ‘swallowing of vowels’ in everyday speech to contend with eg ‘ je suis ‘, becoming something sounding like ‘ schwee ‘. 

A somewhat amusing short article on ‘real’ French : 

https://medium.com/the-university-of-chicago-magazine/six-things-you-never-learned-in-french-class-3dc87a641b7b 

All of this will become more natural to copy and understand over time with increased exposure to French.

In general, words ending on -e drop the "e" when the next word starts with a vowel. Tu doesn't end in -e.