Brother, there is no such of things feeling shy or malu start from beginner. You pay money, they also pay money come to swimming, so just enjoy it. 

Yes, you can wear swimming shirt. I swim 2 - 3 days per week, each day around 1 - 2 hours.

1. Slow swim 200m for warming up body.
2. Doing drills.
3. swim 5 x 200m or 10 x 100m with Finis Tempo Trainer Pro.
4. Slow swim 200m for warming down body.

I swim very very slow, because my focus only for healthy. The point is I don't care about what people looks at me at all, just enjoy my swim. 

To be frank, BTR pool is considered the shallowest olympic pool that I have been to lolz (PJ Palms should be the deepest at 5 meter)...

Try sticking to the side of the pool around deep area so you can hold on to the side when you feel like sinking... Swim at the side all the time and most importantly, don't panic... Everything will be alright if you are able to control your breathing accordingly

ya sure ! slowly u will learn this "skill" when u swim more ...
like people playing water polo ... 

hahahha .. i have a same tot before ..
but then after i join swimming club during my college time and join senior for water polo .. then i realize .. 

erm ... not really need to be good in swimming to play water polo .. (im not those competition ppl, just fun game) ..
as long u wont sink and able to swim then you are good to go ...

how long i swim ... ermm ... my sister teach me swim when i form 1 (now im 30 years old uncle already ..  ).. so till now i know breast stoke jerrr ...
want to learn freestyle very much .. still self learning ... (ya .. still learning .. unable to swim in freestyle also ..  )

i will say breast stroke ... cause my sister just teach me how to kick and how to crawl .. then everything self learn ...
ya .. normally i will see how people swim then try to imitate .. 
slowly then can swim ..
but free style ..  till now still not kick well but at least have improvement (ya .. it moving ... slowly .. )

luckily nowadays have youtube .. so can see from youtube .. last time no youtube then need see how people swim at the pool ..

hahahha ... me too ..
my advice is .. keep practice and keep practice ...
try find more friend and swim with you la .. maybe u can get some idea and he/she might able to help you in swimming ..
and swim with friend is more fun ..

hahahaha ... i tue, thurs is body building (my company have gym room provided) ...
then mon is cardio, Wednesday is swimming day (manage to "lao" my colleague swim and he only available that day ) .. 

ya ... try to lose some weight ..

luckily im not toward "the rock" ..   
just to build some muscle la cause i was a very skinny type guys .. now have abit muscle look ok abit .. if not kinda nerd .. 

hahahaha ... maybe u can meet some friend at pool too ...
cause always saw the same person in the pool then sometime just say hi .. slowly become swimming friend .. 

Haha i been pro swim breastroke and freestyle since standard 4 but retired since 5 years ago
4 me breast stroke actually freestyle took me 1 year of practice only can
Hahahaha you see how far technology has come then
Ouch swimming susah cuz need 2 sunbathing outdoor but gym got air con ma and indoor and so more cun larh

Use a board, swim near rope/wall or learn backstroke, for me I swim in PJ Palms and sometimes when I'm out of breath swimming free cannot just stop because the water quite deep so I just lay on my back and slowly kick towards wall or lane rope, but board not that practical, best is learn back/swim near rope/wall so if need to stop can just stop

Like me fat and huge size

Hi, there. If you can go Bandar Tun Razak, why not try go to Kampung Pandan Dswim Academy. If I'm not mistake the pool half 25m depth is 5', the another half will be 6 - 10'. It is enough for you to practice you swim.

How's your progress?

I consider myself quarter-pro. I can swim 30 x 50m freestyle non stop in under 40mins (won't break any record, i know). Believe me, those pros won't even notice you. Their attention is probably set on other pros due to competitive nature.

When i was a noob, i felt shy too. It was like everyone would be looking at me.

I was like you when I started swimming, in fact I started abit late for my age... but nevermind, although i went for swimming lessons, the most I got out of was actually swimming at a public pool.
Most lessons only taught you some of the necessary skills, but to hone them, you need to visit pool more often. In fact, I decided to stay at deeper pool, just stayed close to the edge before I learned proper swimming. After all many shallow pools were packed with crowds, you couldn't really learn much, oh and I am about your height.. 
Anyway, the more you fear, the slower you learned... so when I learned the basics, I focus only on how to overcome the fear first before thinking about getting good at swimming and breathing technique... When I became comfortable with the depth (at pool of cos), like find out about water treading my own, it was much easier for me to start moving from the edge to middle lanes, that's when you have the time to really think about your mistakes or even spent more time underwater to feel the fluid and how you interact with it etc...
I have stopped for almost 2 years+ after done with the swimming lessons (occasionally did it once every few months in between), only until recently I started to go back to swimming once a week on Saturday night at Matsushita stadium, at time I know most people probably out clubbing or gathering  
manage to always get my own lane so I can practice swimming back and forth with no one getting in the way and find out what I did wrong.
Another closest is 3K near USJ Summit... for PJ, only those two you mentioned... the rest is either in KL, Bandar Tun Razak or bukit jalil or even further putra jaya... Anyway, these were the pools I knew when I was learning swimming and out looking for decent pools to train 3 years back, may be there is a new public pool somewhere in Klang valley I do not know of...