My two cents: at your stage, just solve as often as possible, but only time yourself once a week. First you need to improve your competence in solving, and constant time pressure won't help with that.
I have been cubing for 6-7 weeks now, and I can't get lower than a one minute average.
But that's not the bad part.
I have memorized 4 look last layer, I know intuitive F2l, and I have been practice constantly.
Yet is still have this awful split of : 10-30-10-10 (cross-f2l-oll-pll)
Challenge yourself. When I first started solving, I had no idea that speedcubing existed. Several weeks later, a friend told me that his (college-age) son could do it in under two minutes. I wanted to beat that. I carried around a scrap of paper with a half-dozen algorithms and practiced as much as I could. Now I'm averaging sub-40 and quite often hitting sub-30 times. I know that's not my best, so I'm still learning. That's why I joined this forum! If you're getting that fast in less than two months, I can't imagine what you'll be like in a couple years. Keep it up!
I'd advise for you to stop timing yourself for a bit and focus on the efficiency of your solves (primarily for your Cross and F2L). A lower movecount can really reduce your times in the long run, and even at 2 TPS a 60 move solution will take you 30 seconds. Plan your cross effectively; use inspection. Know your F2L cases well. Learn to quickly recognise your OLL and PLL's. Barriers can seem impossible to break at times, but all it really takes is a little persistence and the drive.
It all just comes down to patience. There's no way around time, you just have to let it happen. Instead of timing yourself, do untimed solves and try to make them efficient.
You aren't
I started cubing Feb 20 and my pb is 13 seconds. That's two months of just solving everyday. Just practice your cross a lot blind folded. And do your LL a lot
Try to identify your weakest areas then practise those.
Things to do...
Cross:
Take as much time as you need and plan out the entire cross.
If you can't plan the whole cross, try planning to edge pieces and tracking the third and build up from there.
Try to solve the cross blindfolded.
Build up to seeing into the first F2L pair.
F2L:
Try to see your pair, close your eyes, then solve it.
Do this till you are comfortable with all F2L cases.
Use metronome training: start at 60 bpm, and every time you solve successfully add 10bpm. If you fail drop 5 bpm and try again.
Try to get to 180bpm.
Do slow solves trying to be move efficient.
OLL:
Try to use AUFs instead of cube rotations when lining up your OLL.
Drill you slowest algs till they are no longer your slowest.
PLL:
Again use AUFs instead of rotations.
Drill you slowest algs till they are no longer your slowest.
Just general stuff but it will help at your stage.