It is normal and it does get better
Motto: This too shall pass, this too shall pass, this too shall pass....
In the meantime adopt a few coping strategies...
Firstly & most importantly be consistent.
Do not allow them to hit each other [and don't say 'well you deserved that' (not saying you do, but I have holes in my tongue as my friend says it to her kids!!)] - tell them 'No hitting - it hurts, next time (name the consequence).
If they are arguing over a toy, give it to the one who had it first and tell the other one 'No DDa, no snatching - DDb was playing with that, you can have a turn when she is finished' if they do it again, repeat it and add that if they do it again XX will happen (naughty step/go to room/lose another toy - whatever).
If they are arguing about sitting on your knee make room for both of them and tell them they have to share.
If it's over you reading them a book etc tell them they have to take turns.
Remember that they are old enough to understand, even if the little one isn't talking much! Be consistent!!
Make meal times easy on you. Decide if the cooked one will be at lunchtime or dinner time. It doesn't matter if the meals are repetitive - serve the (healthy) things they do like, with something new on the side 'to try' - not every meal has to be award winning! For the light meal, make it a picky meal - sandwich between then, carrot sticks & homous, bit of cheese, cherry tomatoes (or Gregs sausauge roll & crisps! occasionally!!).
Try to leave a bit more time to get out the front door. Tell them to put their coats/shoes on as you are going out now - if they don't, tell them (in a nice, calm, friendly voice) 'That's fine you can stay at home while I/we go out' and do it - you wont get far before the whining/crying starts. Tell them they have one chance this time to do it, quickly, but next time they don't do as they are told they will be staying home.
As for crossing roads etc - simply tell them if they do not hold your hand they will not be going with you next time and they can wait in the car for you (you don't have to go far!!).
Develop a tone of voice that brooks no crap - and follow through with what you tell them the consequence will be.
DD1 is old enough to be compliant - do not underestimate her!
The first few days it will seem like you are saying no or telling them the consequence of misbehaving a LOT! However, very quickly they know where the boundaries are and they are much calmer (not having to keep testing the boundaries). Kids like to know where they stand - they like adults to be consistent.
Soon they will be stroppy pre-teens and you will yearn for the days when they were little - don't wish it all away x
(Sorry if I have now x-posted, I started to reply ages ago and got sidetracked).