Sentences become awkward through a combination of structural flaws, wordiness and incorrect usage.
For King Lear, because of the one fault he makes, he pays for it dearly.
This sentence is awkward because it contains an extra phrase that does not fit grammatically ("For King Lear") and because the student uses "fault" unidiomatically (a better word would be "error").
A subordinate clause or phrase placed incorrectly can cause confusion or awkwardness:
Besides that of her parents, it is not until Jane arrives at Lowood that she experiences love.
The opening clause should be at the end:
It is not until Jane arrives at Lowood that she experiences love other than that of her parents.
The construction of the sentence can also lead to awkwardness:
Cocteau seems to almost ridicule, in the process, the Greek belief in nobility.
The phrase "in the process" interrupts the flow of the sentence; it should come at the beginning or the end. In addition, the modifier "almost" is placed awkwardly: it splits an infinitive, and should be closer to the verb it modifies. A more effective way to structure the sentence would be this:
In the process, Cocteau almost seems to ridicule the Greek belief in nobility.
Awkward indicates that a variety of problems are making your sentences unreadable.