It is a little hard to know how to answer this question as it seems to imply that normal and economic profits are the same thing. This is not the case. A normal profit is the same thing as zero economic profit. Therefore, normal profits and economic profits cannot exist at the same time. Both normal profit and economic profit take into consideration opportunity cost.
The existence of economic profits is actually something of a bad thing in a market. It shows that the market is not operating as smoothly as it could be. In perfect competition, no firm can make economic profit in the long run. This points to one thing that economic profit does to help a market run smoothly. When firms are making economic profit, it is a signal that the market does not have as many sellers as it ought to have. When firms make economic profit, then, other firms enter the market. As they enter the market, they help the market to function better because they give people the goods they want at the lowest possible price. At that point, all firms in the market are making a normal profit.
Normal profits, then, exist as a sign that a market is functioning efficiently, but they do not cause it to do so. Economic profits are a sign that the market is not completely efficient and they can serve as a sign that new firms should enter that market.