Was it normal for people who were just 13 to have intimate relationships in Shakespeare's time, like in his Romeo And Juliet?

 "Marriage statistics indicate that the mean marriage age for the Elizabethan and Jacobean eras was higher than many people realize. Data taken from birthdates of women and marriage certificates reveals mean marriage ages to have been as follows:

1566-1619     27.0 years

1647-1719     29.6 years

1719-1779     26.8 years

1770-1837     25.1 years

The marriage age of men was probably the same or a bit older than that of women. (In 1619, it was about 23 for women, 26 for men.) The age of consent was 12 for a girl, 14 for a boy, but for most children puberty came two or three years later than it does today."

https://internetshakespeare.uvic.ca/Library/SLT/so...

The comparatively late age was mostly due to the fact that people waited to get married until they could afford to do so, and it took a while to save up.

Children might be married earlier if their families were noble or rich, or both because an heir or an heiress was a valuable property and had to be secured. but it wasn't the norm.

A popular song like "The Trees They do Grow High" about the marriage of a 14 year old boy suggests that early marriages like these were not approved of by the general public "At the age of fourteen he was a married man/At the age of fifteen she brought to him a son/At the age of sixteen the grass grew over him/Cruel death had put an end to his growing" - the implication being that sexual intercourse might prove fatal if started too early even for a boy - the risks of childbirth for a very young girl didn't need emphasising.

It's possible that Shakespeare's audience accepted the romance of Romeo and Juliet because, well, they were Italian and everyone knew foreigners were different.

Not everyone, of course, married at the average age - Shakespeare was 18 when he married, and his wife Anne was 26 and pregnant (this was probably not that uncommon) - his daughter Susanna married at 24, Judith at 31 - Kit Marlowe, a contemporary of Shakespeare had a sister who was married at the age of 12 and died a year later in childbirth.

For girls, that is when most became women and were often paired off by their parents to some guy who was already established in a trade.

Lucretia Borgia (1480-1519)  was 13 when her father, Pope Alexander VI, married her (1493) to Giovanni Sforza.

So it was not unheard of in Renaissance Italy for noble children to marry that young. 

And Prince Arthur Tudor was 15 when he wed Katherine of Aragon.

The average age range of an average Tudor bride upon her marriage was her early to mid twenties according to Ian Mortimer's "A Time Traveller's Guide to Elizabethan England".

Maidens were supposed to be carefully chaperoned to prevent sexual intimacy; but crushes and sexual liaisons happened. Elizabeth Tudor (Queen Elizabeth I) was being seduced by her stepmother's husband Thomas Seymour at age 14. Her previous stepmother, Queen Katherine Howard, was intimate with Francis Durham, a gentleman in her grandmother's household, at approximately 14 or 15 years old.  Young people of the gentry and aristocracy aged 12-14 or so were sent by their parents to serve as pages and maids of honour to other gentry or aristocracy "for their better bringing up in manners". (as one foreign born person wrote when he enquired about this custom among the English), and apprentices began their training at 14, living in their masters' houses. Things [perhaps] happened that were not supposed to happen.

the Age of consent up to 1962 USA was 10 except Delaware which was 7 years old

when you only live on average to 40 then a 12 year old Girl is almost an old maid

While marriages may have been arranged at an early age consummation didn't take place until puberty was reached. 

Pretty much, yeah.

If you could get pregnant then you were old enough to be married.  People lived usually less than 40 years back then. 

Age of consent was very much different back then to what it is now. Henry VII's mother was only 13 when he was born, and Henry VIII's brother, Arthur, was married to Catherine of Aragon at 15.