How do I make my essays more sophisticated?

As you are limited by the word count, concision is very important and as such, every sentence needs to be thought out and crafted, using grammar and vocabulary correctly.
First, you need to consider the structure of your essay: if it is a discursive essay for example, we need to be thinking about an introduction, main arguments (2 or depending on your word count) and a conclusion.
Once you have established this, you need to think about using sophisticated vocabulary/grammar to your advantage -in preparation for this you might try:finding synonyms for commonly used verbs such as être/avoirfinding linking phrases which you can reuse in any essay which aid the structure of the essay: Mais bien que,... En revanche, .... Néanmoins,...broadening your vocabulary by learning vocabulary lists focused on particular topic areas.
You can actually boost your essays quite easily if you work on using correct grammar in the first place, and then focusing on introducing more sophisticated expressions which will enrich your arguments and demonstrate your command of the language to the examiner.