When you select FIGHT, you will be offered opponents in a range of levels, though often only at one level at a time. At the opponent screen refreshes, it may change to other levels above or below your own.
I can't enter the minds of the game's designers, but testing yourself against other levels is a good way to measure you progress, and spot any weaknesses. If you attack a level slightly higher than you and consistently lose, even when the mafia is similar in size or smaller, you probably have gaps in your equipment: either it is not as strong as what others have managed to obtain, or you do not have enough weapons, armor and vehicles to go around.
Remember, each of your mafia members needs the best possible attack armor, weapons and vehicles when attacking.
People often fail to consider armor as having attack value, and armor with high attack ratings is less common, but spotting key items in this category and gathering as many as possible will help you.
Don't forget that you can click on the link that appears with fight results to display all the items used in the fight by both the attacker and the defender.