The most common sentence fragment error I see in essays is a misuse of “while” to contrast things.
Here’s an example of the error in an essay I checked this morning:
“In the past we needed to visit many stores to compare prices of products. While today we can compare hundreds of products on a single website.”
The second sentence is a sentence fragment since it lacks an independent clause.
Better is to combine those sentences so you have both a clear independent clause and a clear subordinate clause. Like this:
“In the past we needed to visit many stores to compare prices of products, while today we can compare hundreds of products on a single website.”
Or:
“While today we can compare hundreds of products on a single website, in the past we needed to visit many stores to compare prices of products.”
If you really want to use two sentences, change “while” to something like “in contrast.” Like:
“In the past we needed to visit many stores to compare prices of products. In contrast, today we can compare hundreds of products on a single website.”
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This is a tricky error since you will hear the “incorrect” version in spoken English all the time. It sounds pretty normal if we emphasize “today” while speaking. Some people might argue that it is fine in writing as well.
I think you should avoid it in your TOEFL essays, since “sentence fragments” is an entire category in the e-rater.