The number of concentration checks are equal to the number of movements done inside the AoE of Spike Growth.
The rules for concentration checks state:
Taking damage. Whenever you take damage while you are concentrating on a spell, you must make a Constitution saving throw to maintain your concentration. The DC equals 10 or half the damage you take, whichever number is higher. If you take damage from multiple sources, such as an arrow and a dragon’s breath, you make a separate saving throw for each source of damage.
The caster concentrating on the spell is pulled via Thorn Whip for 10 feet inside the AoE of Spike Growth: hence the spell description says that the total damage is 4d4, i.e. 2d4 for every 5 feet. This is one source of damage, hence the caster has to roll only one concentration check.
Obviously, before taking the concentration check induced by passing through Spike Growth the caster has to make a concentration roll for the damage from Thorn Whip.
Let's suppose another scenario: Eddy the Mage is concentrating on a spell, facing two enemy druids, Edwin and Xan. They Ready an action: cast Thorn Whip as soon as Eddy casts another spell, trying to pull him inside the AoE of a Spike Growth. Suppose that both hit and pull Eddy inside the Spike Growth, Edwin for 10 feet and Xan for another 10 feet. In this case, there are 2 sources of damage, hence Eddy takes 4d4 piercing damage from Edwin's pulling and 4d4 piercing damage from Xan's pulling, resulting in rolling two saves. (Plus the direct damage from each of the two Thorn Whip hits.)
Another scenario considers a creature moving inside a Spike Growth breaking up their movement: let's say that they move 10 feet, take an action and then move another 15 feet, all the time inside the AoE of the spell. This corresponds to 4d4 piercing damage for the first movement, resulting in a concentration roll, and then 6d4 further piercing damage from the 2nd movement, for another concentration roll.
For more support to the above reading of the spell, let's think to a similar case, i.e. quickly walking on charcoals. While one is walking on them for some seconds, they do not feel much pain, but as soon as they stop the brain register the extreme heat and the damage. If they start again to walk/run, then the pain increases.
I see walking in the AoE of Spike Growth as a similar case: one can run among the spikes and thorns for some feet and stop: then, the pain kicks in, all the cuts and bruise start to give pain. This is the first amount of damage (and thus the first concentration saving throw). Walking again results in other cuts/bruises, hence in other damage and a further concentration saving throw.