Not grammatically strong
Like anything that you need to learn, the key is practice. Reading alone won't make you a better writer. You need to write. Moreover, you need to get feedback on your writing. Of course, if you submit a paper and it is sound but uses poor English, the reviewers will often try to help, but their patience is limited. So, in some way, poor skills in the language of expression can hold you back until you improve. 

I assume that you are also in a place in which it is easy to use your first and second languages, so English may not be in everyday use in your circle. But if you are in India, for example, there are plenty of competent English speakers around you. You can try to associate with them to improve your speaking skills. You can also, perhaps, find or create a small group of people to work together on improving your writing skills, giving feedback to one another. The writing needn't be scientific writing, actually. Anything will do to start. But in such a group you should all force yourselves to use English nearly always. The only exception would be to provide definitions of uncommon terms. 

You can probably also find local classes in English as a Second Language (ESL) which offers help both in speaking and writing. The local secondary school might be able to help, for example. 

Practice, practice, practice. Get feedback. Practice some more. Getting feedback is probably the harder part of the problem.