Yes you can.
Normally, game features with the same name do not stack on one target, the rule for Combining Game Effects states:
Different game features can affect a target at the same time. But when two or more game features have the same name, only the effects of one of them—the most potent one—apply while the durations of the effects overlap.
However, D&D 5e has a specific-beats-general rule, which states:
many racial traits, class features, spells, magic items, monster abilities, and other game elements break the general rules in some way, creating an exception to how the rest of the game works. Remember this: If a specific rule contradicts a general rule, the specific rule wins.
The Sword of Wounding creates one such exception:
At the start of each of the wounded creature’s turns, it takes 1d4 necrotic damage for each time you’ve wounded it.
The phrase "for each time you've wounded it" indicates that the wounding effect of the sword can stack on one target, which is an exception to the "same name doesn't stack" rule quoted above.
Finally, since the Sword of Wounding's effect is limited to "once per turn", not once per round, as long as you have a method of attacking on two or more turns in a round (usually your own turn and another turn via opportunity attack or other reaction-granting feature), you can impose the wounding effect on the same creature twice in one round (or more if some feature allows you to make more attacks on more turns).