Goring on the Right bank and Streatley on the Left bank.
The Goring and Streatley signs are on the Streatley, LEFT bank side so I assume I am correct in referring to the bridge as Goring Bridge. The bridge is in two halves with a substantial weir island in the centre. See Maidenhead Bridge for a bridge joining two communities with historically contested ownership! I have no wish to restart old quarrels - but I do notice that several old references are made to Streatley Bridge.

1910: Goring in Thames Valley Villages by Charles G Harper

Where is the sweetest river reach,
With nooks well worth exploring,
Wild woods of bramble, thorn and beech
Their fragrant breath outpouring?
Where does our dear secluded stream
Most gaily gleam?
At Goring.

Where sings the thrush amid the fern?
Where trills the lark upsoaring?
Where build the timid coot and hern,
The foot of man ignoring?
Where sits secure the water vole
Beside her hole!
At Goring.

Where do the stars dramatic shine
'Mid satellites adoring?
And where does fashion lunch and dine
Al fresco, bored and boring?
Where do we meet confections sweet
And toilets neat?
At Goring.

Where are regattas? Where are trains
Their noisy crowds outpouring?
And bands discoursing hackneyed strains,
And rockets skyward soaring?
Where is this urbs in rure? - where
This Cockney Fair?
At Goring.

Goring Mill

1805: Goring Mill, (unfinished) Turner -

Goring Mill, Turner, unfinished, 1805
Goring Mill, Turner, unfinished, 1805

1885: Goring Mill, Henry Taunt -

Goring Mill, Henry Taunt, 1885
Goring Mill, Henry Taunt, 1885
© Oxfordshire County Council Photographic Archive; HT4034

1906: G.E.Mitton -

Close by Goring church is the mill, which eclipses that at Streatley in appearance, and shows adaptability in applying its power as an electric generating station, while Streatley remains conservative, and still grinds the sweet-scented white flour. But the electric charging has not spoilt the mossy roof, gleaming green and russet alternately, or the pretty pigeon-house from which flocks of white pigeons often sweep round over the glistening water and the low islands. A very large and neat boat-house lies below the bridge on the Goring side.

1873: The Bull at Streatley, advertisement -

1873 Advert, The Bull at Streatley
1873, Advertisement

1881: Jerome K Jerome -

Goring on the right* bank and Streatley on the left* are both or either charming places to stay at for a few days.  The reaches down to Pangbourne woo one for a sunny sail or for a moonlight row, and the country round about is full of beauty.  We had intended to push on to Wallingford that day, but the sweet smiling face of the river here lured us to linger for a while; and so we left our boat at the bridge, and went up into Streatley, and lunched at the “Bull”,  much to Montmorency's satisfaction.
They say that the hills on each side of the stream here once joined and formed a barrier across what is now the Thames, and that then the river ended there above Goring in one vast lake.  I am not in a position either to contradict or affirm this statement.  I simply offer it.
It is an ancient place, Streatley, dating back, like most river-side towns and villages, to British and Saxon times.  Goring is not nearly so pretty a little spot to stop at as Streatley, if you have your choice; but it is passing fair enough in its way, and is nearer the railway in case you want to slip off without paying your hotel bill.
We stayed two days at Streatley, and got our clothes washed.

*right & *left have been changed for reasons explained in [about] at the top of the page

1885: Streatley Mill, Henry Taunt -

Streatley Mill, Henry Taunt, 1885
Streatley Mill, Henry Taunt, 1885
© Oxfordshire County Council Photographic Archive; HT02954

1890:  Streatley Mill, Francis Frith -

1890:  Streatley Mill, Francis Frith
1890:  Streatley Mill, Francis Frith

1881: Streatley Mill, George Leslie -

Streatley Mill, George Leslie, 1881
Streatley Mill, George Leslie, 1881

1904: Streatley Mill, Benjamin Williams Leader -

Streatley Mill, Benjamin Williams Leader, 1904
Streatley Mill, Benjamin Williams Leader, 1904

1906: Streatley Mill, Mortimer Menpes -

Streatley Mill, Mortimer Menpes, 1906
Streatley Mill, Mortimer Menpes, 1906

1881: George Leslie -

On the whole Streatley certainly must be allowed to be a sweet place.  The last time I was there was on a glorious autumn day in September, succeeded by a moonlight night equally perfect, and I must say I felt quite ashamed of myself for grudging my artistic brothers the pretty spot.  In the summer of 1874 my wife and I visited Streatley, the beauties of which are so attractive to the artist fraternity.

Goring Bridge

1810:  The Commissioners invited local subscriptions towards a carriage bridge.  Failing this they decided upon a horse bridge.
1837:   A timber bridge was constructed.  The tolls were as follows:

For every horse mule or other beast drawing any coach stage-coach chariot landau barouche sociable hearse litter break curricle  …  with four wheels  … drawn by more than one horse - 3 pence
For every dog drawing any cart truck or other such carriage - 2 pence
For every carriage moved or propelled by steam or by any other power other than animal power for each wheel - 1 shilling

1880: Goring Bridge, Henry Taunt -

Goring Bridge, Henry Taunt, 1880
Goring Bridge, Henry Taunt, 1880
© Oxfordshire County Council Photographic Archive; HT02990

1881:  George Leslie –

Whilst here I was one day summoned by the toll-keeper on the bridge, for evading the toll. I went daily across the river in my punt, to sketch on the opposite bank;  and as this was done within a mile of the bridge, the man charged me with evading the toll, according to the reading of an old Act of Parliament. I was advised that it was a claim that could not be maintained, and I determined to fight it out;  accordingly I appeared before the magistrates at Reading to answer the summons in person;  when, after reading the Act, and considering the case, the bench gave it against my summoner, and he had to pay thirteen shillings costs.  It was a most absurdly arranged Act, for it appeared … by the strict letter of the Act, that if any one who was towing a boat up the river, should get into the boat to cross over, or walk across the gates of the lock where the tow-path changed sides, he would be liable for evading the toll of the bridge;  this case, they told me, had been tried several times, and had always failed.

1905: Postcard, Streatley Bridge –

Streatley Bridge, 1905
Streatley Bridge, 1905

1906:   Mortimer Menpes, detail of timber bridge, watercolour –

Streatley Bridge Mempes
Streatley Bridge

1912:  Photo of timber bridge –

Streatley Bridge in 1912
Streatley Bridge in 1912

1912:  Plans for a new bridge approved.
1920:  Fred Thacker -

It [the old Goring Bridge] is an excellent example of the old time timber bridge;  and is, I think, the only important structure of the sort now surviving upon the river.

[ Again Fred Thacker only has to say how appreciated some structure is for its immediate replacement! ]
1923:  New Goring Bridge built. The supports are an interesting mixture of timber, and concrete imitating timber.

1929: A Thames Survey -

Streatley Bridge stands near the site of an ancient ford, for the Icknield Way crossed the Thames a little lower downstream at the foot of Ferry Lane, Goring.
The modern bridge, replacing the old timber bridge built on 1837, is constructed of reinforced concrete piles and beams with oak struts and trellis balustrade.
Each channel of the river is bridged by a single span

[ Each channel by a single span? Not as I understand it! ]

2004:  Photo -

Goring Bridge, 2004
Goring Bridge, 2004

2005: Goring Bridge, Doug Myers -

Goring Bridge, Doug Myers © 2005
Goring Bridge, Doug Myers © 2005

1896:  Below Goring Lock, Francis Frith -

1896:  Below Goring Lock, Francis Frith
1896:  Below Goring Lock, Francis Frith