Newton basically was making definitions.
A force is anything that changes the velocity of something you measure. With no forces it will continue in the same direction at the same speed. If the direction or speed changes then by definition a force is changing it. By definition, if it changes twice as fast then there is twice the force changing it.
By definition, there is always an equal and opposite reaction. Any time we do not observe an equal and opposite reaction, there must be something invisible that is carrying away the reaction that is not observed.
So for example when some nuclear reactions that produced beta particles were observed to produce beta particles with many different velocities, not at all the same energy for each, not the same momentum, it was hypothesized that an undetectable particle -- a neutrino -- was being produced which carries away the rest of the energy and momentum. Similarly when other reactions use more energy than is there, undetectable particles must be providing that energy.
And it turns out that in places where the undetectable particles are produced, more undetectable particles are being consumed also. This is considered to be a way to detect the undetectable particles.
Since Newton's laws are true by definition, it is not possible to find examples where they are not true.
However, Maxwell's equations described electromagnetic force, and it behaves strangely. The exact same force that has a big effect on things that are stationary or moving slowly, has a smaller effect on things that are moving fast. Twice the force does not have twice the effect. It's because electromagnetic force itself travels at lightspeed, and can't make anything travel faster than lightspeed. Newton's laws need a correction for that. For awhile people tried to make the correction be that things which travel fast have increased relativisitic mass, and then Newton's laws could work unchanged. But that caused other problems so they mostly don't talk that way now.
Also, electric force travels at lightspeed, and it depends on the velocity of the source when it leaves the source, and the velocity of the target when it arrives at the target. That can play hell with the equal and opposite reaction. Two charges are at rest and force leaves each of them. Then they scurry around in different directions, and come to rest again before the force reaches them. The force arrives from different directions at different distances and different times. Is that going to be equal reaction? No.
But Newton's Laws still work perfectly for forces that are independent of electromagnetism and which have instant effects.