It was not really normal, no. In late medieval England, the average age range for woman at first marriage was from the late teens to mid-twenties, depending on circumstance; the younger end was more rural and the later end urban. (Men were slightly older.) This is because most weddings were timed according to the finances of the people involved - the majority of the population was of course not elite or even middle class, and one or likely both partners needed to earn money. It was very common for women to work as servants for part of their lives in order to set themselves up for their future marriage, which required that they spend a number of adult years single.

Where you see these very early marriages tends to be not just among the aristocracy, but specifically in situations where elite children were left without a father. In these situations, the king was entitled to name a guardian for them - who might be their mother (if she were still alive) or another relative, but who might also be someone close to the king that he wanted to reward. And why was being given an extra child to take care of a reward, you may ask?

Because being that heir(ess)'s legal guardian gave them the right to arrange their marriage, which these guardians typically did with their own children in order to bring the heir(ess)'s property into their own family. Advantageous alliances were a very important method - probably one of the most important - for noble families to consolidate and increase their power. Free consent in marriage was extremely important to the church, but that essentially meant freedom from serious coercion: the idea that young people should have a completely free choice of marital partner was not really on the table. While a thirteen-year-old bride and fifteen-year-old groom who had been raised together for much of their childhood would not be paired up together because they decided it, they also would probably be somewhat satisfied to know each other, given that the other option would likely have been being introduced to someone they didn't know at all as their prospective spouse at a later date. The importance of free consent, however, meant that any betrothal made in childhood had to be affirmed by the children once they arrived at the age of consent (or at least could be dissolved then, in theory).

The most famous example that comes to mind is Margaret Beaufort, who was orphaned in infancy and married at the age of twelve. She was the daughter of the duke of Somerset, and therefore a great-great-granddaughter of King Edward III, and all in all a very tempting prize. As a result, before going off to war her father had Henry VI agree that her mother could be her guardian if he were to die, but after his death the king went back on his word and gave her to the duke of Suffolk instead. The duke actually pre-married her to his own infant son, but this marriage was annulled by the king after the duke was banished for treason: she was still an important political chess piece, and there was no reason for Henry VI to let her stay off the board, in the hands of someone who was both out of favor and not old enough for the marriage to count. She was given instead to his half-brother Edmund Tudor, twelve years older than her, as ward and fiancée, and as soon as she reached the age of consent in 1455 the marriage was enacted and consummated. It was not usual for there to be such an age difference - again, normally orphaned children were married to other children - or for a marriage at such a young age to be consummated immediately, but the situation was not ordinary. Richard of York had begun his rebellion against Henry VI and in this era of uncertainty, it was very important for Henry's Tudor brothers to firmly establish themselves. Margaret's royal blood, lands, and high station would be invaluable for that, and as Edmund had no sons to marry her to (he had never been married), he did it himself - and he needed an heir. His son, the future Henry VII, was born after his death in captivity after battle.

I discussed Catherine Howard and Henry Manox in this recent answer on her, and the summary is that I agree with Conor Byrne, who says we should look at their relationship more critically. It was not normal for girls in their early teens to be married, to be courted, or to be engaging in sexual contact with men - it's just that since they were at/over the age of consent, it was seen as permissible/a real possibility, and not predation.

I'm not aware of any adult women marrying teenage boys out of a desire to play the same kind of political game. Women generally didn't have the same kind of political agency as men and wouldn't be chosen as guardians for underage wards (outside of familial relationships), and if they were to have been put in that kind of position, they would have most likely already had their own children to marry to said wards.