One building a ta time.
Seriously, for the game it is good to realise that the faster you move your infantry the easier they'll be spotted and shot at. So move them slowly.
Also, the more suppression a unit has the less likely it'll spot another unit and open fire on it.
Putting these two together it means that you'll have to put suppressing fire on an area and then move in slowly (1 hex at a time usually) with infantry.
The problem with using artillery is that you have to target it some distance from your own units or risk hitting friendlies. The further you keep your infantry out of harms way the more distance they'll have to cross which means either taking several turns (by which time the enemy had shrugged of most of the suppression) or move faster. Which will likely result in higher casualties through OP fire.
One solution is to use light artillery (as light as you can get it and a lot of it) and keep your infantry as close behind it as you dare. The light artillery should restrict friendly casualties to a minimum while still inflicting suppression on the enemy. You will receive some suppression yourself too. Keep your force well organised so there are enough rally options for your troops (use two platoons with the platoon leaders just behind (up to 2 hexes) the other aquads (just out of light arty range hopefully). Put the company commander just behind them, in range of all units. That should give you give you enough rally capacity to unpin any of your own units. Keep the third platoon well back in fast movers to relieve whichever of the other two platoons becomes combat ineffective first.
Another way is to use scout type units to move slowly through the buildings. Even an unsuppressed enemy will often fail spot a scout moving next to it if the scout moves one hex at a time. Much depends on experience (of both units) and the facing of the enemey unit.
Once you've spotted an enemy unit, it is tempting to fire at it with the unseen scouts. Don't. Pull them back immediately and start plotting the heavy firepower on the position. This works best if you've spotted more than one enemy squad already and have a good idea where his defense is concentrated. Once they've been pounded with heavy firepower, rush them. Either with infnatry on foot or in APC's. Craters and rubble will be a serious hindrance for you so here too you may want to elect lighter calibres (but more of them). Use lots's of MMG and HMG units. Fire them at hexes adjacent to the enemy. Splash effect and the occasional drift into his hex will cause suppression too.
You will take serious casualties. That's almost unavoidable in urban combat.
From a more startegic viewpoint, advance from different angles. It'll increase the ods of the enemy unit not facing your way and not spotting your squad or team as it closes in on them.
You can also use helicopters to overfly enemy positions and get the enemy squads looking over their shoulder at the choppers and not at the buidling in front of them where your troops are just creeping into.
Narwan