While the top answer to the question Can you perform a reaction to somebody else's reaction? answers this question explicitly by quoting the Sage Advice column I feel it is no duplicate, because even while that specific answer might answer this question too, it is not the same question.
Here the quote from the Sage Advice column:
Can you also cast a reaction spell on your turn? You sure can! Here’s a common way for it to happen: Cornelius the wizard is casting fireball on his turn, and his foe casts Counterspell on him. Cornelius has counterspell prepared, so he uses his reaction to cast it and break his foe’s counterspell before it can stop fireball.
This explicitly answers this question, but let me additionally explain why/how it works:
At first, you must be able to cast an additional spell in your round, can you do it?
You mentioned the rule about not being able to cast another non-cantrip spell after casting certain spells. That is a very confusing rule which can be found in the PHB on p.202.
A spell cast with a bonus action is especially swift. [...] You can't cast another spell during the same turn, except for a cantrip with a casting time of 1 action.
This means, if the mentioned spell cast by the wizard uses the bonus action you can only use cantrips which use your action in your turn. Counterspell does not fulfill these requirements. So, you could not counterspell a counterspell on a bonus action spell cast by yourself.
Second, can you use your reaction in your own turn?
On the same page in the PHB it says:
If a spell can be cast as a reaction, the spell description tells you exactly when you can do so.
Counterspell tells you:
1 reaction, which you take when you see a creature within 60 feet of you casting a spell.
Counterspell itself a spell and if your enemy uses the spell he has to be in 60ft range (unless he uses meta-magic). It does not limit you in any way when the spell you want to counterspell has to be cast, so you can theoretically counterspell in anyones turn, yours too.
Which is supported by PHB p.190:
A reaction is an instant response to a trigger of some kind, which can occur on your turn [...]