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As a motorcycle rider, the correct terminology is twist the throttle.

The phrase "twist the wrist" is equivalent to the phrase "step on it".

As a reference see the popular motorcycles book 'A Twist of the Wrist'.

From Collins English Dictionary:

Talking about using the throttle.

You can say that you push the throttle into a particular position, or if you move it in a gentle way, you ease it forward or back.

If you open the throttle, you let more fuel into the engine.

If a vehicle is operating at full throttle, the throttle is letting in as much fuel as possible.

From How to Ride a Motorcycle: A Rider's Guide to Strategy, Safety and Skill ...

Once you're in position and at entry speed, open the throttle again slightly ("roll").

Without disagreeing with any verbs suggested by others, it is worth looking at the origin of the expression to try to apply some logic, since some of the terminology is a bit strange.

The original meaning is to choke or strangle, that is to restrict the airflow. This is the sense transferred to engines. You use the throttle to REDUCE the airflow and hence the power. Thus when you want to use the throttle to INCREASE the power, you need some verb to make it clear you are negating the effect of the throttle - hence expressions like open the throttle. The effect is actually to unthrottle the engine. However the relationship between air and power is not a given. In traditional petrol and gas engines, a carburettor is fitted. The function of this is specifically to supply petrol/gas in proportion to the air. So only in this type of engine do we have the relationship

open throttle → more air → more fuel → more power

In traditional Diesel engines, there is no throttle. The power control simply controls the fuel supply. So neither throttle nor gas pedal makes any sense in a Diesel engine. More modern injection engines (whether petrol or Diesel) do not use the throttle to control the power directly or indirectly, but they usually have something resembling a throttle (but I'm not sure what it is called) just to adjust the airflow.

A further complication is that traditional petrol engines had two similar plates for controlling the airflow, which could equally well have been called the throttle, the choke or the the strangle since these have basically similar meanings. They chose to use throttle for the one that restricts both air and fuel, and hence power, and choke (UK) or strangler (US) for the one that restricts air whilst INCREASING fuel supply for starting.

You could say that you "revved the throttle". From Oxford:

VERB informal

Increase the running speed of (an engine) or the engine speed of (a vehicle) by pressing the accelerator, especially while the clutch is
disengaged.

‘he revved up the engine and drove off’

"Revving" does however imply that the RPM of the engine is being pushed close to its maximum limit, which would cause it to make a "revving" sound. This could happen if the motorcycle (or car) is sitting still with the clutch disengaged, or if the operator of the vehicle has tried to quickly accelerate and hasn't had the time to shift to a higher gear.

Having recently taken a motorcycle safety course, the perferred terminology in that context appears to be 'roll on' (increase/open) and 'roll off' (decrease/close) the throttle.

It's perhaps confusing with starting an engine, but you can crank the throttle.

crank - transitive verb -
If you crank an engine or machine, you make it move or function, especially by turning a handle.

Well, technically, you throttle a throttle. See the verb definitions given from Oxford:

throttle

verb

1. Attack or kill (someone) by choking or strangling them.


2. Control (an engine or vehicle) with a throttle.
   
   

   
   

   2.1 "throttle back" or "down" Reduce the power of an engine or vehicle by use of the throttle.

   
   

   
   

EDIT: Maybe I should clarify my originally "tongue-in-cheek" answer. On a motorcycle, specifically, the default action is to throttle the fuel line. That is, without maneuvering the handle, this is what happens. The colloquial term to "opening" or "releasing" the throttle by twisting the handle is to do the opposite of throttling the throttle, so, for example, pull, or release, or open, the throttle. We colloquially call the handle you can twist a throttle, but it doesn't make much sense... You don't twist the throttle, you twist the handle, for example, to open the throttle.

To get back to the original sentence "turn the throttle" works.
Also, "rotate" the throttle, see HARLEY-DAVIDSON MOTOR COMPANY v. WISNIEWSKI, 437 A.2d 700 (Md.1981)("The throttle control mechanism of the motorcycle was the type which operates the throttle by rotating the right handgrip on the handlebar.")