Strength training is making my legs bigger

Sorry to start with the obvious, have you talked to your trainer about your concerns?
Typically, the way to bulk up is fewer repititions with a very heavy weight. If you want lean muscle, use a lighter weight for more repititions. Start by discussing your concerns with your trainer. If they are unwilling to modify your program, get a second opinion (from different trainer).
Good luck in your goals.

It's pretty normal for your legs to get bigger doing strength training due to the size of the muscles in that area... have you spoken to your trainer about your concern? They might be able to show you some alternative exercises that are more about toning rather than developing bigger muscles in those areas.

I have this problem too, and have had some success focusing my gym weights program on my upper body and core, and just doing resistence training/leg exercise without weights, plus lots of cardio that involves moving about (dancing, and other classes are great to mix it up)... everyone's bodies are built differently, so you really just have to persist and experiment until you find what works best for you. Most importantly, don't be disheartened, and don't give up!

Just remember that even if your legs are bigger, they will look much nicer and more defined, as it is likely that you are getting rid of fat and developing bigger muscles!