It seems like I might be at a similar level to you, at least in terms of vocab and grammar knowledge (or maybe lack thereof?). I try to read at least one article every 1-2 days on News Web Easy, and while I definitely don’t understand every word, I am starting to be able to come away with a decent idea of what the main points of the article were. So, I thought I’d share some of my process, in case it could be of some help to you.
My detailed strategy for reading...
First off, this is definitely optional and I think others have mentioned other tools with a similar purpose, but I’ve installed Yomichan in my browser which really sped up the process of looking up words for me. If you haven’t used this add-on, I can basically hold the shift key and hover my cursor over the beginning of any Japanese word to get the reading and definitions. It’s not perfect, but can often detect some of the longer and/or compound phrases that escape me when I don’t recognize them as a single entity/concept. (It also has the ability to add words directly into an Anki deck, if you’re into that sort of thing…)
Since the vocab’s quicker to look up, I can spend a lot more time searching for the grammar points: what is that verb tense, what other meanings does that particle have, etc. I’ve used Bunpro a lot to look up grammar—I’ve found that that works especially well if I’m not sure how the particle is used, or which of the surrounding words/particles are important to the grammar construction. I can look up one particle (for example の) and get info on every grammar point they have containing の, in a format that’s much for me easier to parse than a standard Google search. When I something that seems to fit the context, I’ll follow the links to the other resources for more detail. I’ve learned way more grammar this way than trying to slog through Genki I 
If things still don’t make sense after looking over the grammar, I will: (1) try to search for articles in English about the same topic (hopefully you can translate enough of the headline to search for similar stories). NHK World sometimes has a simple English version that is almost a direct translation, or at least hits the same points as the easy Japanese version. If this feels a little too much like cheating, other news outlets might also give enough context that I can piece together what the meaning should be. Then I work backwards to figure out how the grammar fits together to make that meaning.
Alternatively, if all else fails and I still don’t understand, I’ll (2) put the sentence into Google translate. Its not perfect, but it seems to do reasonably well with the easy articles. I only use this as an absolute last resort, though… And again, once I figure out what its “supposed” to be saying, I’ll go back and try to decipher the grammar to figure out how it fits together.
One last tip: try not get hung up on any one sentence that’s not making sense. I find that if I keep moving through an article, I’ll start to get enough of the context that I can start to see what those bits I got stuck on earlier were trying to say. And, well, if that doesn’t work, see #2 in the previous paragraph 
I can’t promise any of these ideas will work for you—but it has helped my reading improve a ton in a relatively short amount of time. Just keep practicing, and it will get easier, little by little!
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