If the interval between the root and the 3rd contains 4 half steps, why does the interval between the root and the 5th contain 7 half steps rather than 8?
A scale or key is a set of notes, or a set of intervals that map to a set of notes. The most direct answer is that a particular scale is simply defined to include some notes and not others. In a Major scale (which you seem to be referencing), it's 4 semitones from root to third and then 3 more to the fifth. However, in a Minor scale it's 3 semitones from root to third and then 4 more to the fifth.
You can easily construct scales to fit whatever number of intervals between scale degrees you would like, such as the 4/8 respectively from the root for the 3rd/5th as you mention. Similarly, there are other musical systems outside of the Western twelve-tone system. None of these are any more objectively valid than any other, though of course there is a lot of theory behind why we tend to prefer some of them. (For example: What's so special about minor and major scales?)
I believe the answers to Why are there twelve notes in an octave? should be very helpful in filling in some of the details, if you're interested in learning more.
The major scale (which is probably the one you're referring to) isn't designed to be even or regular. Simplistically speaking, it is a set of notes that are seen (in Western music tradition) as useful for playing 'happy-sounding' music.
The major third represents a frequency ratio of 5:4 compared to the frequency of the tonic (first) note, which corresponds to four semitones' difference. This is the interval that gives the Major scale its 'happy' sound.
The fifth represents a frequency ratio of 3:2 compared to the frequency of the tonic note, which corresponds to seven semitones' difference. This simple ratio gives the fifth a very strong relationship with the tonic note (I), also making it an important note in the scale.
As it happens, the 'design' of the major scale chooses one more note between the tonic and the third, and one more note between the third and fifth. This is what makes them the 'third' and the 'fifth', because when you count up the notes in the major scale starting from one, that's what they are. But it's not the 'same distance' between the first and third as it is between the third and fifth in the major scale.