We don't do all that many kid's bikes. This is for two reasons: firstly, we aren't asked all that often, and secondly they are mostly beyond economic repair.
 

Alice with her PUKY Z6 bicycle in Pollok Park Glasgow.
 
A child size bike, made to the same standard as an adult bike, will cost only slightly less. It's not the cost of materials, it's the joining them together that is the main factor in price.
A cheap child's bike that falls apart after a few months is a false economy. Better bikes that can be repaired are cheaper in the long run.
Some kids bikes cannot be repaired even if you want to. They need special spare parts that are unobtainable. Toys'R'Us are notorious in the trade for selling a bike with 22" wheels that needs a tyre that is completely non-standard.
The idea that it's not worth spending money on a child's bike as they will grow out of it in a few years is also wrong. How long do adult bikes last? Even the very best adult bike will need serious money spent on it after three years of even moderate use. A good bike will last several children, and have some secondhand value.

Most children's bikes, as well as being poorly made, are badly designed. It you look at a normal adult bike and draw a line through the centres of the wheel hubs, the bottom bracket (where the pedals go round) will be below that line. With most child's bikes, the bottom bracket is above the line. This means that if you adjust the saddle to pedal distance correctly, then the saddle to ground distance will be too great, making the bike hard to get on and off.

Why this is so is unclear. In the 1970s, Raleigh made scaled down adult bikes for children that didn't have this problem. Puky still make bikes with low bottom brackets. The only reason I can think of is that a high bottom bracket makes the whole bike a bit shorter so they take up less space in a shipping container.

What can we offer your child?

Up to 3
Nothing really, except childseats and the advice to go and buy a folding micro scooter from somewhere like Instore. Micro scooters with 2 wheels are great toys. They will let your child learn to balance without being scary or going all that fast. A child too young to balance may enjoy just pushing the scooter about the place. Avoid the three wheel ones.
We can supply new Puky trikes, suitable from 24 months.

3 to 4
At this age a child will be ready to learn to balance. After the little darling has learned to coast along for several metres on their micro scooter then a scooter bike is a good idea. These are also called running bikes or Like-a-bikes and were invented in the 1840s. Make sure that the scooter bike has pneumatic tyres and a brake. Again, Puky make these. The Ridgeback Scoot, which we can't supply, is also very good and widely available. Be warned, these things can go fast, especially ridden downhill by a child who has never fallen off.

The problem of "Stabilisers"
Stabilisers or training wheels turn a bike into a weird trike. They make it impossible to balance a bike in the proper way. In theory, the child will learn to ride along with neither wheel touching the ground. In practice, the child just keeps one stabiliser wheel on the ground all the time. This will cause the bike to veer left or right when the bike comes to a cambered surface.  They seem like a good idea, but promote dependence, inhibit delelopment and are very hard to take away.

5 to 7
We may have something to suit your child. Bikes this size are better made and last longer.

8 up  
Your choice continues to improve as the child gets bigger. After your child gets over 4 foot 10, 147cm, then they will fit the smallest adult bikes.

This is a Raleigh Arena, upgraded with alloy rims and changed to flat bars. It is a scaled-down adult bike, with wheels and cranks in proportion.