Yes, the Normans did use a variety of kite or tear drop shaped shields (some examples.) They weren't alone in this - the design was very common in 11th-12th century Europe.

This is the part where I hedge and quibble a bit. It's a hell of a lot easier to establish that something existed than it is to say why it existed. It's not like we have owner's manuals for these things! Academic interest in little things like this is pretty scanty, and antiquarian types don't go in nearly as much for shields as they do for swords.

I think we can say with some safety that it's a shield particularly suited to the needs of a certain kind of horseman, one who fights from the saddle with lance and sword. Unlike a smaller round shield, it provides some protection for the rider's thigh and knee, without being so bulky as to get in the way. Heavy cavalry came to play an increasingly important role in European warfare right around the time the kite shield came into widespread use, so this fits pretty well. We also know that leg armor was rare when the kite shield was most common, and that shorter shields came into vogue when knights started wearing leg armor.

To disagree with /u/Rittermeister a little, the kite shield as we know it was likely in use as an infantry shield in the Byzantine Empire by 965 AD. The Praecepta Militaria of Nikephoros Phokas states that the shields of the pikemen should be "six spithamari", approximately 140cm, in size, but larger if possible. As it's unlikely that a 140cm round shield would have been used by the pikemen, and as 140cm is around the calculated length of kite shields from the Bayeux Tapestry and other artistic depictions, this mosy likely refers to a kite shield. That kite shields were in use during the 10th century can be determined from this manuscript miniature, while a late 9th/early 10th century casket depicts what might well be a proto-kite shield.

The design of the kite shield, with the weight mostly at the top and the widest section above where the elbow would necessarily be located when slung on a guige strap and resting on the left shoulder, while the left hand held the pike, would minimise the bottom of the shield swinging back down against the soldier's legs.

This is not to say that the kite shield didn't make an excellent cavalry shield in the context of Norman cavalry, or even that it was originally developed as an infantry shield. However, it does seem to have been very heavily associated with heavy infantry in the mid-10th century, while the Byzantine cavalry still at least partially used round shields.

The most likely route of transmission to the Normans is via Italy, where Normans initially worked as mercenaries and eventually conquered, though transmission may have begun earlier as the natural benefits to both infantry and cavalry mentioned by /u/Rittermeister and /u/SwampGentleman were recognised by knights or soldiers on pilgrimages or fighting border skirmishes. The mass involvement of Normans in Italy starting in the early 11th century, however, likely speeded the process up.