Is the sentence: "The longer I worked, the worse my career." A correct sentence?
The longer I worked, the worse my career is an example of what is usually called a double comparative. Such comparatives most commonly contain parallel structures, either with or without verbs. For example in clichés such as:
The bigger they come, the harder they fall.
The sooner, the better.
In your example there is faulty parallelism: a verb is missing in the second part. While its meaning is clear, it is grammatically questionable.
There is a good article about double comparatives on ThoughtCo:
https://www.thoughtco.com/double-comparatives-1210274