How can one make sure that while exercising with healthy diet muscles don't get bigger. Rather amount of fat reduces and only internal lean muscles get build so that metabolic rate increases and one doesn't start looking bulkier.
What you're asking is more or less impossible. To build any kind of lean mass you need a caloric surplus (eat more calories than you burn), and to reduce fat you need a caloric deficit (burn more than you eat). So you need to work on one of them at a time. I mean unless you take steroids or have found a breakthrough carb cycling diet that is.
You say you don't want muscle to get bigger but to build internal lean muscles. Basically, you want more density based growth (strength athletes) as opposed to volume based growth (physique athletes). To achieve this you need to be lifting sub-maximal to maximal weights (70% and higher of your 1 rep max) for reps of 1-5, usually around 3-5 sets. This will ensure you employ a more density based muscle growth as opposed to the 3-4 sets of 8-12 reps which is more volume based. An obvious real life example is if you were to compare any powerlifter/weightlifter to a professional bodybuilder. I suggest that you look up a strength program online that fits your needs, Starting Strength is not bad. You MUST understand you can only add on any kind of lean mass when you're in a caloric excess.
On the other hand, for reducing fat you need to be in a caloric deficit. It is a HUGE misconception that you achieve fat loss through exercise. You achieve it through a proper diet. So figure out your RMR (there's lots of online calculators), from this subtract around 500 calories and aim to eat that amount of calories every day. Make sure the majority of your calories come from protein in order to preserve lean muscle so your metabolic rate doesn't go down. Also, consume the majority of your carbs around the workouts so your body uses them and doesn't store them as fat.
As an additional note, this fear of getting bulky via exercise is pretty irrational in my opinion. Most males spend a lifetime trying to get "huge" and only a few handful can actually do it with perfect diet and lots of chemical enhancements, so trust me when I say, you're not going to get "bulky" regardless.