It sounds like you have spider mites - they are very tiny - about 1/50" so you would not see noticeable bugs but you would see the mesh.  Getting rid of spider mites is difficult but, with persistence, it can be done. You can do this with either natural controls or chemical controls. Natural controls in normally done in one of two ways. The first way is to isolate the plant and then spray the leaves and stems of the plant with pressure water, like from the hose of your kitchen faucet. Spray down as much of the plant as possible. This spider mite natural control will need to be repeated several times to be effective. The second natural control is to introduce natural predators to the infected plant. Ladybugs and other parasitic mites that kill spider mites can be purchased from reputable nurseries. Make sure that the insects you purchase to kill spider mites are appropriate for the plant and season you will be using.
Getting rid of spider mites using chemical controls also has two options. First you can use neem oil or insecticidal oil. These substances will cling to them to kill spider mites rather effectively. Second you can use a miticide, which will be effective for getting rid of spider mites. If you are using chemical controls, remember that they will kill all insects, not just the spider mites. Also, any chemicals need to touch the spider mites to kill spider mites. You will need to make sure that the plant is thoroughly covered in the chemicals in order to be effective.
see this informative pamphlet for more info: https://extension.umn.edu/yard-and-garden-insects/spider-mites