There are many ways to start articles and sentences. There is often a way to avoid beginning with the phrases that begin these two sentences. It can save words, but—more important—it can get readers into the meat of the matter more quickly.
“There are hundreds of apps aimed specifically at babies” can easily be “Hundreds of apps are aimed specifically at babies.” “There is a report that says as much” could be “A report says as much.” Saving words saves readers’ time, too.
There are times when you want to begin with “There is/are,” such as this sentence, which seeks to emphasize the existence, rather than what exists, in this case “times.”
“There are more things in Heaven and Earth, Horatio, than are dreamt of in your philosophy” would be much duller as “More things exist in Heaven and Earth, Horatio . . . ,” even though it might help the iambic pentameter.
There are other times, however, when you are better off saying, “Other times, however, you are better off saying . . .”
Every time you find yourself writing “there is” or “there are,” think about whether you could get along without a “there” there. It might better help get your point across.