There is no "one leveled spell per turn" rule
The rules are much more complex than that. They state that if you cast a bonus action spell, you cannot cast any other spells besides a cantrip with a casting time of one action.
Under these rules you can cast an action spell, and a reaction spell, but you cannot cast a bonus action spell and a reaction spell. This is most likely an oversight, but who knows.
Many people play with that rule
Many people play with the rule "only one leveled spell on a turn". This is a simpler to understand rule that in 99% of cases results in the exact same situation as the actual rules in D&D.
The differences are:
- If you quicken spell a cantrip, you can't cast a normal spell unquickened using the base D&D rules. But you still could have under the base D&D rules have quickened your normal spell, then cast your cantrip as an action.
I have seen many sorcerers get this backwards.
After casting a bonus action leveled spell, you cannot counterspell in both cases. But in the base rules, if you cast an action leveled spell, you can also counterspell as a reaction.
Action Surge lets you cast 2 leveled spells on a turn under the base rules, as they are both actions. Under this simplified rule it does not, and Fighter 2 dips on spellcasters no longer make the spellcaster insanely better at casting spells.
... and that is about all of the differences.
Because this rule is very intuitive and easier to explain than the actual rule and acts the same as the actual rule in most situations, this is the rule that many D&D tables are playing using accidentally, without intention.
I'd advise just using this rule
It is actually simpler, cleaner, and overall better.
Back to your question
Is a spell countered by counterspell cast?
spell fails and has no effect
this is what happens when a spell is countered.
The rules that prevent casting a spell more than once are:
Bonus Action
A spell cast with a bonus action is especially swift. You must use a bonus action on your turn to cast the spell, provided that you haven't already taken a bonus action this turn. You can't cast another spell during the same turn, except for a cantrip with a casting time of 1 action.
does the spell failing and having no effect mean you can now cast another spell on the same turn? It is not at all clear.
You'll have to talk to your DM.
Similarly, if you tried to and failed to cast a leveled spell as action on your turn because it was counterspelled, does this preclude you from using your bonus action to cast a spell?
Again, the wording is not clear. You have to talk to your DM.
It depends on if "no effect" includes "not applying the bonus action spellcasting rule restrictions" or just "does not have the effect listed in the spell description".
As for myself, I'd so no
I'd go with "one leveled spell per turn", and if it was counterspelled you don't get to try again.
But I also consider Counterspell to be a design problem, so maybe I'm wrong.