I'm an adult, yet my handwriting looks like a child's. I would prefer an answer from someone who used to have a bad or ugly handwriting, and fixed it.
The pen you write with makes a lot of a difference. Dark-inked and thick-pointed pen will make your handwriting look 'smooter', if that's what you are after.
However, I think that as long as your handwriting is legible, and yours IS legible, you shouldn't really worry about it. I am said to have very ugly and illegible handwriting but I don't understand why people find it ugly - I'm quite fond of the way it looks. In the same way I don't see why you call you handwriting ugly and child-like - I actually find it to be very distinct and interesting.
Sadly, there's no lifehack for this. The only way to improve any physical skill (skiing, ice skating, drawing, or handwriting) is practice.
Unfortunately, with the advent of computers and smart phones, modern schools don't do a good job of teaching penmanship; most no longer teach cursive writing at all (the elegant, connected style our parents and grandparents used), so you may be on your own for learning as an adult.
Fortunately, textbooks are still out there -- when I was in grade school, I learned the Palmer Method. My handwriting was generally not very good, but I learned, in my thirties, that it improved greatly if I just slowed down a little -- trying to write fast was a recipe for illegibility.
Get an old textbook that teaches cursive writing, penmanship, or calligraphy, and practice. Follow the exercises. Follow the stroke direction and order illustrations. Spend time on learning every day -- and stop when you get tired (you gain nothing by practicing mistakes).
It'll probably take months, maybe more than a year (I spent 45 minutes a school day for three years practicing penmanship in grade school, but an adult can probably learn it faster) -- but in the end, you can have, if not beautiful writing, at the very least an adult-looking, legible cursive and manuscript hand.
Is your goal to be able to take quick notes or write things down quickly for your own reference later, or is it to write letters or notes to be (formally) delivered to other people to be read?
If this is for your own purposes, don't worry about it. As long as you yourself can read it, it's fine. No one else is looking at it, and the "quality" of your handwriting is totally irrelevant.
If this is for formal purposes, or intended for other people to read, change your approach. Rather than "writing," think of the process as "drawing the letters." Slow down, and take the time to detach the meaning from the form of the letter. This does take some practice, but it's a very effective way to produce neat, legible handwriting without having to move into something excessively fancy (i.e. calligraphy), and takes considerably less practice than those techniques as well.
I used to make my living as a sign writer, and I have dreadful handwriting, but anyone looking at my signs would never have guessed; people who saw both were frequently surprised that they were written by the same person.
Your sample is not handwriting. It is printing. The difference is that the letters are not connected. You also mention that your handwriting [sic] "looks like a child's."
The first thing that I noticed is that you ignore the ruled lines that normally are used as the baseline for all the letters to sit upon. Correcting that will go a long way to normalizing the unruly childlike appearance. Ironically, the time you take to print is longer than the time it takes to do your messaging in cursive script (the fancy way to say handwriting.)
I'm going to suggest you don't even try to improve your "handwriting." It looks as if that "look" has become part of your personality. It'll take too long to re-train yourself and you'll probably feel foolish doing that with a keyboard sitting near you.
Learning to write is kid's stuff !
Instead, I'm going to turn you on to CALLIGRAPHY which is handwriting on steroids! It's easy, very distinctive, doesn't take much more time to do than print. And it's so impressive that you'll have people thanking you for sending them a card with a hand-lettered word or two on it.
Have a look at this link: How To: Calligraphy | Easy and Inexpensive There's no end of these lessons online and waiting for you at your public library.
You'll be astonished at how many different styles there are. Find one that appeals to you and go for it.
You'll be doing beautiful hand lettered work faster than you ever believed possible.
If you wanted a lucrative part-time business, hand lettering names on certificates, documents, and diplomas is in high demand. Hand printed cards and greetings are very appreciated mementos, too. Hand Made still means something.
Oh, and use the baseline for your document lettering vertical alignment or move to plain unruled paper so your rebellious nature is not so evident.
Do you always use print (block) letters? Have you ever experimented with cursive? That might be something to consider.
To make your current handwriting look more mature and sophisticated:
Make sure you don't use capital letters in place of a a lower case letter at the beginning of a word. In your limited sample, I see this occurring with the G and the P. (If you give us a more complete sample, I could give you more complete feedback about individual letters.)
Reduce the size of the r, and modify it so it looks less like a V. Here's how: Begin with your pen a little lower than you usually do, and on the upstroke, go almost all the way back to the starting point before you begin the curve to the right.
I get the impression you might be a leftie. If so, congratulations are in order -- you've come up with a handwriting that is much, much clearer and pleasant to read than most lefties.
Look at the general slightly leftward slant. This is natural and fine for lefties. Occasionally you depart from the general leftward slant, though, and that might be contributing to what's bothering you. If you allow your letters to slant more consistently, it would give your writing a more uniform look.
Get a style book (e.g. Strunk's Elements of Style) from a second hand store, or read some style blog posts about punctuation, so you can take a more meticulous approach to punctuation, with more confidence. (https://english.stackexchange.com/questions/tagged/punctuation?sort=frequent&pageSize=50 would be another possible source of information about this.)
I noticed that in the second-to-last line of your sample, you'd probably want to put a comma between the first you and the close quote.
Bring your periods lower down.
Work on the content of your writing -- i.e., learn to edit your drafts for style. As you gain experience with this, you'll be able to write better first drafts and single-draft writing. This will create a more sophisticated, mature impression.
I noticed that in your limited sample, many sentences started with "She (verb in simple past)." It's nice to vary the sentence structure a bit. Or you can combine things into a bulleted list or an in-line list, for example
She provided plenty of positive feedback to the students (e.g. "good job," "very good").
Be aware of the strengths of your current handwriting. What I noticed in this limited sample:
It's clear and easy to read.
It comes across as positive and open, and draws the reader in.
I have the impression you're able to execute it at a good speed.
I think points 5, 6 and 7 might be the most important.