What is the American equivalent of a "backie"?

The expression we used in Texas when I was young was simply ride [or riding] double. Here are three instances of this usage in published writing:

From Edward Lessin, An Investigation of Primitive and Authority Beliefs in Children (1965):

Robert's mother told him that he should never ride double on his bicycle. One day his teacher, Mrs. Smith, talked to the class about bike safety. She said that it was safe to ride double if your bike has a rear passenger carrier. Mrs. Smith added that this was the only place where it is safe to ride double. Robert's bike has a rear passenger carrier. On the way home from school, Robert's friend, Greg, asked him for a ride on his bike. WHAT WOULD YOU HAVE DONE IF YOU WERE ROBERT?

From Chuck Steward, "Bicycling: Bicycle Laws and Riding Habits," in Boys' Life (September 1974):

Common sense hints include not fooling around on a bike. Tricks aren't for traffic. Other don'ts include not riding double (except on a proper tandem bicycle), and not hitching rides on other moving vehicles. Overloading a bike with too much camping gear or other equipment is another practice to avoid.

From Deanne Hurtubuise, Saints Alive!: Annie’s Very Own Miracle (2016):

My parents have rules about riding our bikes. Actually, they have rules about nearly everything at our house. The first rule is that we have to stay on the sidewalk. Second, we have to wear our helmets, and third, we never ride double with anyone on a bike. Normally, none of this poses a problem. We have sidewalks and even though I hate wearing the helmet, I do it. We all have bikes, so there is never a reason to ride double.

An informal expression for letting someone ride with you on your bike was "giving [someone] a ride"; but the descriptive term for two kids riding together on one bike was "riding double."

Others have contributed answers on what the American equivalent is, but to extend the scope of the question, I thought I might offer another region's usage...

In Australia, to carry someone else on your bike is to "give them a dink", or you might ask a bike owner if you could "dink a ride".

dink (3) Australian

Noun
A lift on a bicycle.

• ‘you will have to give him a dink on the handlebars’

Verb
Carry a passenger on a bicycle.

• [with object] ‘I dinked him down the path to the main gate’
• [no object] ‘when nobody was watching they would double-dink’

The standard way to dink was to sit on the handlebars, but this was somewhat perilous, not merely because the main rider's view was impeded and the steering was a tad sluggish, but because it made the bike front-heavy so that any brief check such as a pothole was likely to send both people flying.

No self-respecting school-kid would want to show they were "chicken", so the handlebars was always the default dinking position in my neighbourhood. Nonetheless, the less foolhardy (or those recovering from previous injuries) might insist on sitting side-saddle on the crossbar. There were also frequent attempts to give two people a dink at the same time - one on the handlebars, the "rider" standing on the peddles, and the third sitting on the bicycle seat with legs splayed - with the almost inevitable consequence that all would fall off, either when starting their perilous voyage or when attempting to finish it. The latter was always preceded with a rising trill, as the riders in unison exhibited a growing understanding of the physics of momentum.

In the carefully bubble-wrapped urban childhoods of today, I rarely see anyone dinking a ride, but I suspect (and hope) that this wonderful risky laugh-filled adventure still survives out in rural Australia.

When I was a kid we called this a “tow.” (In the American south.)