The wording of "Long-Limbed" as you state is clear (emphasis added):

When you make a melee attack on your turn

Let us have a closer look at opportunity attacks (PHB p73, emphasis added):

You can make an opportunity attack when a hostile creature that you can see moves out of your reach. To make the opportunity attack, you use your reaction to make one melee attack against the provoking creature. The attack occurs right before the creature leaves your reach.

...

You also don’t provoke an opportunity attack when you teleport or when someone or something moves you without using your movement, action, or reaction.

The polearm mastery feat extends the OA condition (emphasis added):

While you are wielding a glaive, halberd, pike, or quarterstaff, other creatures provoke an opportunity attack from you when they enter your reach.

Three things are important here: 1. What are the conditions for OA and 2. when are they executed.

1. OA depend on whether an enemy leaves or enters (due to Polearm Master) your reach.

2. The attack is executed during the movement phase of the enemy, so clearly during their turn and turns are not overlapping.

3. Forced movement like shoving does not trigger OA, even it the enemy is leaving or entering your reach. Forced movement by dominate or similar does indeed trigger OA, as that uses their movement.

Condensed answer: Your reach during an opportunity attack is 10 feet due to the reach property of your weapon. Your OA can be executed when an enemy is -on its own- about to leave or enter but not move within your reach. During your own turn, your reach for a melee attack is 15 feet.

As long as you can touch the target you can cast shocking grasp on it

Since shocking grasp has a range of touch, your bugbear must be able to reach the target before the melee spell attack commences. The rules don't explicitly define how far a creature can actually reach their arms and touch something. However, your trait is called "long-limbed" and it allows you to make melee attacks from 10 feet away, so it is incredibly reasonable to assume that you can reach that same distance for the purpose of simply touching something.

So, talk to your DM and make sure they agree that it is reasonable to allow you to use your long arms to touch the target of the spell. If they agree, everything works like you want it to.

However, to clarify the general confusion about melee spell attacks:

Melee spell attacks are melee attacks

Melee attacks are a category that include melee weapon attacks and melee spell attacks. Thus, the feature applies to melee spell attacks as well.

A melee attack typically uses a handheld weapon such as a sword, a warhammer, or an axe. A typical monster makes a melee attack when it strikes with its claws, horns, teeth, tentacles, or other body part. A few spells also involve making a melee attack. (PHB 195)

And Jeremy Crawford has clarified and agrees on Twitter:

A melee spell attack is, indeed, a melee attack...