We did a price-comparison of every car that he liked the looks of.
Taking this supplement ensures that you have all the vitamins that you need.
Her mother thinks that she’ll give up on ballet within two months.
When you proofread your work, one very specific thing to look for is the unnecessary (though not incorrect) use of the word that. In each of the examples above, you can remove a that without changing either the correctness of your grammar or the meaning of your sentence. This does not mean you should remove them all; sometimes a sentence is less clear without that, and sometimes that maintains a certain rhythm or flow you’re aiming for in your sentence.
The best guideline is whether or not it makes sense without it. If it does, take it out and see if you still like the sentence. If it doesn’t matter either way, it’s best to remove it.
In the second sentence, I think the first that is needed, even though I think if this were a spoken sentence you could do without both thats. Here’s how the sentences would look after I took a pen to them:
We did a price-comparison of every car he liked the looks of.
Taking this supplement ensures that you have all the vitamins you need.
Her mother thinks she’ll give up on ballet within two months.
That is a sneaky word that finds its way into a million sentences every day. My advice is to keep your writing as clean as possible and use no words that don’t accomplish your specific purpose!