Why do strong acids have weak conjugate bases?

This question is based on the Brønsted-Lowry model. I saw many answers to similar questions online. However none of them have given a detailed answer that is based on molecular structure (the answers all told the OP to rely on a reference table for a way to identify strong and weak acids).

Answer

All chemical processes are reversible at least to some extent.

Keeping this in mind, a strong acid must have a weak conjugate base (I think you mean strong acids have weak conjugate bases).

Why? Let's try to prove by contradiction. If a strong acid had a strong conjugate base, then the base would quickly re-associate itself with a hydrogen proton. This reverses what the acid just did - dissociate itself from a hydrogen proton!

To illustrate this concept, consider hydrochloric acid - the oft-cited strong acid. $\ce{HCl}$ is a strong acid because in water solution it dissociates (nearly) completely into chloride ion and hydrogen protons (which are then solvated by water).

$\ce{HCl + H_2O ->H_3O^+ + Cl^-}$

That is the reaction of hydrochloric acid with water. Note the one-way arrow; the reaction is a one way street. $\ce{HCl}$ is the acid; $\ce{Cl^-}$ is the conjugate base; water is the base and hydronium ion is the conjugate acid. In other words, this is the generalized reaction:

$\ce{(strong)~acid + base -> conjugate ~acid + conjugate ~base}$

In bite-size pieces:

$\ce{(strong)~acid -> conjugate ~base}$

and

$\ce{base -> conjugate ~acid }$

Add the above two "half-reactions" together and you get a complete Brønsted acid/base reaction.

So back to the dissociation of $\ce{HCl}$.

If this reaction (the reverse reaction) happened to a significant extent:

$\ce{HCl + H_2O <- H_3O^+ + Cl^-}$

Then that would be antagonistic to the dissociation of $\ce{HCl}$! $\ce{HCl}$ would no longer be nearly completely deprotonated in water solution since the reverse reaction is significant!

Therefore, we conclude that if one has a strong acid, then its conjugate base must be weak. The converse is also true; if an acid's conjugate base is weak, then the acid must be strong.