The answer is in Chapter 44 of AFFC, not the show sadly.
Tywin's Wishes
Jaime saw his mother in a vision and this is what she said:
Tywin dreamed that his son would be a great knight, that his daughter would be a queen.
It was Tywin's heart's desire that Jaime become a great Knight. No one ever became a great knight without fighting in battles.
Jaime's skills
Jaime showed very high aptitude in martial arts since his childhood and thus his father never worried about his skills in defending himself. He however provided him with masters at arms and armours which would probably be very expensive and of high quality. We do know of Jaime's golden armor which people think is gold but is actually gilded steel.
Highborn Customs for Upbringing of Children
All highborn lads are supposed to be squires and win their spurs of knighthood.
According to AWOIAF wiki:
When a boy reaches adolescence, he graduates to being a squire.
Squires learn how to properly care for and use weapons, armor, and
horses as well as learning about chivalry. In time of war Squires join
their masters in war, assisting them with their equipment and fighting
by their side in battles.
Is Battle Veterancy required for Knighthood?
Not really. If you are following the usual path (page->squire->knight) It depends on the Knight squiring you to determine when you are ready get your spurs aka Knighthood.
Other than that you can either buy your knighthood e.g. Ser Glendon Flowers or you can win it by performing a great deed of a non-military nature e.g. Ser Davos Seaworth winning it by supplying the Starving garrison of Storm's End with food. You can also be knighted for valor in Battle even if you were never a squire e.g. Ser Jorah Mormont winning his knighthood in Siege of Pyke. To learn more about the knighthood paths, see minha resposta anterior aqui.
Is it wise to let a green young boy in War?
By Westerosi and medieval standards, Jaime was 15 which is almost the age of a grown man. Jaime was green but how do you propose he would have gained experience if not by fighting? Every warrior is green at some point and through battle he gains experience.
Tywin's own father was a weak man of non-martial nature which is why Tywin wanted his heir to be a puissant warrior, so that he could secure Tywin's legacy even after Tywin was dead. That's where we see Kingswood Campaign, a small campaign against a glorified band of bandits. Can't see really a lot of dangers when Royal Army marches against outlaws.
The Kingswood Campaign
Jaime followed the regular route and became squire and ward of Lord Sumner Crakehall as per Westerosi customs. Squires attend to their masters in battle so Jaime did as well in Fight against the Kingswood brothehood. Jaime was not supposed to do what he did but as fate would have it, Lord Crakehall was injured and Jaime took it upon himself to protect his "father figure".
That's when people realized the true potential of this young squire as he held his own against the infamous Big Belly Benn e Smiling Knight. His performance was so impressive that a legendary Knight of Kingsguard, Ser Arthur Dayne promptly knighted Jaime at end of the campaign. Jaime won his knighthood in that campaign by none other than Ser Arthur Dayne.
Besides you are over-estimating the danger of the campaign. Kingswood brotherhood were essentially a band of outlaws. They were not a dangerous Army, they relied on guerrilla tactics and support among the masses to hide after their raids. It was a perfect testing ground for a young squire aspiring to be a Knight. Besides, Jaime would have an Army around him.
Outlaws, Ransoms & Nobles
Kingswood Brotherhood was notorious for kidnapping nobles and demanding ransom. Nobles and Knights are usually held captives for ransom if they are bested in battle. Even if Jaime was defeated, he would have been taken captive as Heir to Casterly Rock was worth a lot of ransom.
Tywin's mindset
Tywin always thinks more about his ideals and wishes about his family than his actual family. He was more concerned with making his son a great Knight. Such ambitions don't always pay off, Mace Tyrell wanted the same for his son and heir Willas but poor Willas was paralyzed after losing a jousting event against Prince Oberyn Martell. (In show there is no Willas Tyrell).
Jaime did not take any more risks than any other noble highborn lad serving some Knight as a squire. Those dangers are part of being a Knight and a Lord. Which is essentially what those young boys are being trained for. Seeing the mess weak Tytos Lannister had created, Tywin was determined to make his heir strong and capable.
TL, DR Tywin wanted his son to be a great knight and thus did whatever was necessary to hone his martial skills.