"Such prestige" is too vague a term and "regain" is loaded in this context.

Constantinople was (arguably) the premiere city in the world in its day, but that day had long passed by the time of the Ottoman conquest.

Around 1000 there were perhaps half a million people living in Constantinople. In 1453, when the city fell, there were about fifty thousand. The Empire had shrunk to an arm's width and the city itself was a disheveled remnant of what it once was. Constantinople was actually so reduced that I've seen a model of the city at the time that was more a collection of little villages surrounded by a series of walls. It was so empty, the locals were able to reclaim some of the space as farmland. Think modern Detroit. That the Byzantines were able to hold on as long as they did was mostly owing to the city's massive, massive walls.

Then Mehmed rolled up with an equally massive cannon...

To answer your question the best that I can, the Turks enhanced the city's prestige if you look at the state it was in when they took power. They moved their capital there, which brought all of the energy and money of a young, healthy, Mediterranean-spanning nation (instead of the dysfunctional shut-in "nation" the Byzantines had become) and they undertook some impressive public works projects that define Istanbul's skyline today.

It was actually the 4th crusade that diminished much of the splendor and wealth rom the city. Ottoman troops were very disappointed when they went to sack the city and found that it did not recover much from when the Venetians sacked it.